| L(s) = 1 | + 2·5-s − 9·9-s + 44·13-s + 16·17-s − 59·25-s + 34·29-s − 12·37-s + 136·41-s − 18·45-s + 178·53-s − 100·61-s + 88·65-s + 72·73-s + 54·81-s + 32·85-s − 404·89-s + 266·97-s + 628·101-s + 664·109-s + 208·113-s − 396·117-s + 499·121-s + 86·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2/5·5-s − 9-s + 3.38·13-s + 0.941·17-s − 2.35·25-s + 1.17·29-s − 0.324·37-s + 3.31·41-s − 2/5·45-s + 3.35·53-s − 1.63·61-s + 1.35·65-s + 0.986·73-s + 2/3·81-s + 0.376·85-s − 4.53·89-s + 2.74·97-s + 6.21·101-s + 6.09·109-s + 1.84·113-s − 3.38·117-s + 4.12·121-s + 0.687·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(38.25366007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(38.25366007\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{3} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - T + 31 T^{2} - 134 T^{3} + 31 p^{2} T^{4} - p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( 1 - 499 T^{2} + 122219 T^{4} - 18507970 T^{6} + 122219 p^{4} T^{8} - 499 p^{8} T^{10} + p^{12} T^{12} \) |
| 13 | \( ( 1 - 22 T + 332 T^{2} - 3124 T^{3} + 332 p^{2} T^{4} - 22 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( ( 1 - 8 T + 331 T^{2} - 112 T^{3} + 331 p^{2} T^{4} - 8 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( 1 - 904 T^{2} + 326888 T^{4} - 93346162 T^{6} + 326888 p^{4} T^{8} - 904 p^{8} T^{10} + p^{12} T^{12} \) |
| 23 | \( 1 - 406 T^{2} - 226321 T^{4} + 236989580 T^{6} - 226321 p^{4} T^{8} - 406 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( ( 1 - 17 T + 859 T^{2} + 7358 T^{3} + 859 p^{2} T^{4} - 17 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 31 | \( 1 - 2229 T^{2} + 2639094 T^{4} - 2726532929 T^{6} + 2639094 p^{4} T^{8} - 2229 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( ( 1 + 6 T + 2676 T^{2} + 2896 T^{3} + 2676 p^{2} T^{4} + 6 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( ( 1 - 68 T + 3847 T^{2} - 191464 T^{3} + 3847 p^{2} T^{4} - 68 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 43 | \( 1 - 7084 T^{2} + 25586768 T^{4} - 58817360782 T^{6} + 25586768 p^{4} T^{8} - 7084 p^{8} T^{10} + p^{12} T^{12} \) |
| 47 | \( 1 - 4626 T^{2} + 17180271 T^{4} - 44845226876 T^{6} + 17180271 p^{4} T^{8} - 4626 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( ( 1 - 89 T + 11023 T^{2} - 524698 T^{3} + 11023 p^{2} T^{4} - 89 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 59 | \( 1 - 19987 T^{2} + 169266515 T^{4} - 778415071426 T^{6} + 169266515 p^{4} T^{8} - 19987 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 + 50 T + 3015 T^{2} + 100444 T^{3} + 3015 p^{2} T^{4} + 50 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( 1 - 13212 T^{2} + 116453088 T^{4} - 609293414990 T^{6} + 116453088 p^{4} T^{8} - 13212 p^{8} T^{10} + p^{12} T^{12} \) |
| 71 | \( 1 - 24082 T^{2} + 262463183 T^{4} - 1677810300412 T^{6} + 262463183 p^{4} T^{8} - 24082 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 36 T + 14976 T^{2} - 357446 T^{3} + 14976 p^{2} T^{4} - 36 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 - 15669 T^{2} + 159095862 T^{4} - 1251251095745 T^{6} + 159095862 p^{4} T^{8} - 15669 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( 1 - 30211 T^{2} + 434597339 T^{4} - 3751922129410 T^{6} + 434597339 p^{4} T^{8} - 30211 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( ( 1 + 202 T + 9763 T^{2} - 84460 T^{3} + 9763 p^{2} T^{4} + 202 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 97 | \( ( 1 - 133 T + 32363 T^{2} - 2538046 T^{3} + 32363 p^{2} T^{4} - 133 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.58902760373661077804570758698, −4.21747817032927388077896822625, −4.20653627030241443262847909887, −3.98340364297180381892253643317, −3.78812408499577277345176097870, −3.75577284635871712820318685244, −3.56936463780131744392360948804, −3.28248343018026584316069504171, −3.20268967287359855451966227342, −3.15733105847355516936778805201, −3.12395939536049963118006245431, −2.80355195315064861859534741822, −2.63461072263113455503786702936, −2.26798585667955653521584352069, −2.13037294872580287193357945641, −1.94402639987399047956257179826, −1.90967057874488339725234941458, −1.80315300344870719487294398105, −1.63922709018389039331800610505, −1.05617901390012885842373942602, −0.989936261671606997457969006240, −0.75748201756925353124771287753, −0.71261901814365175810052210706, −0.49689986967358769074835421372, −0.47306943238210990691327938114,
0.47306943238210990691327938114, 0.49689986967358769074835421372, 0.71261901814365175810052210706, 0.75748201756925353124771287753, 0.989936261671606997457969006240, 1.05617901390012885842373942602, 1.63922709018389039331800610505, 1.80315300344870719487294398105, 1.90967057874488339725234941458, 1.94402639987399047956257179826, 2.13037294872580287193357945641, 2.26798585667955653521584352069, 2.63461072263113455503786702936, 2.80355195315064861859534741822, 3.12395939536049963118006245431, 3.15733105847355516936778805201, 3.20268967287359855451966227342, 3.28248343018026584316069504171, 3.56936463780131744392360948804, 3.75577284635871712820318685244, 3.78812408499577277345176097870, 3.98340364297180381892253643317, 4.20653627030241443262847909887, 4.21747817032927388077896822625, 4.58902760373661077804570758698
Plot not available for L-functions of degree greater than 10.
