| L(s) = 1 | + 18·5-s + 7·9-s − 64·13-s + 14·17-s + 131·25-s + 128·29-s − 2·37-s − 160·41-s + 126·45-s + 94·49-s + 14·53-s + 158·61-s − 1.15e3·65-s − 286·73-s + 81·81-s + 252·85-s + 194·89-s − 352·97-s − 30·101-s + 226·109-s − 448·117-s − 233·121-s + 342·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 18/5·5-s + 7/9·9-s − 4.92·13-s + 0.823·17-s + 5.23·25-s + 4.41·29-s − 0.0540·37-s − 3.90·41-s + 14/5·45-s + 1.91·49-s + 0.264·53-s + 2.59·61-s − 17.7·65-s − 3.91·73-s + 81-s + 2.96·85-s + 2.17·89-s − 3.62·97-s − 0.297·101-s + 2.07·109-s − 3.82·117-s − 1.92·121-s + 2.73·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.545922051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.545922051\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 94 T^{2} + p^{4} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 7 T^{2} - 32 T^{4} - 7 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 9 T + 56 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 233 T^{2} + 39648 T^{4} + 233 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T - 240 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )( 1 + 647 T^{2} + p^{4} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 697 T^{2} + 205968 T^{4} + 697 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 1801 T^{2} + 2320080 T^{4} + 1801 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + T - 1368 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 2098 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 2807 T^{2} + 2999568 T^{4} - 2807 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 7 T - 2760 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 4153 T^{2} + 5130048 T^{4} + 4153 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 79 T + 2520 T^{2} - 79 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 8857 T^{2} + 58295328 T^{4} + 8857 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 7778 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2}( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 11257 T^{2} + 87769968 T^{4} + 11257 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 13714 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 97 T + 1488 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 88 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.032339177978438087498718079872, −7.25049481133839713907164713491, −7.18787991142440947446010668210, −7.05946966802863971181015050224, −6.87099201473158836969679817910, −6.51369258399595986517994019719, −6.47366202922867516322483691172, −5.96681358950092257998367867366, −5.67674933390745789131474786436, −5.48225022119763886318896234176, −5.40578599409973778717773110376, −4.88753146553633232563436635702, −4.80665827181817146335910321998, −4.79510165968560233979055911569, −4.40181999976631460083834061115, −3.92589253053704072683227497190, −3.30532309726928598219755974303, −2.87427947598787424818679799056, −2.69499679851895316424253689376, −2.48559893247186060354692811743, −2.20156947223036813898365000328, −1.85550699909573195294462944956, −1.62897550637810705587238324268, −1.03791526467424555354298566892, −0.40843593749708722065815372032,
0.40843593749708722065815372032, 1.03791526467424555354298566892, 1.62897550637810705587238324268, 1.85550699909573195294462944956, 2.20156947223036813898365000328, 2.48559893247186060354692811743, 2.69499679851895316424253689376, 2.87427947598787424818679799056, 3.30532309726928598219755974303, 3.92589253053704072683227497190, 4.40181999976631460083834061115, 4.79510165968560233979055911569, 4.80665827181817146335910321998, 4.88753146553633232563436635702, 5.40578599409973778717773110376, 5.48225022119763886318896234176, 5.67674933390745789131474786436, 5.96681358950092257998367867366, 6.47366202922867516322483691172, 6.51369258399595986517994019719, 6.87099201473158836969679817910, 7.05946966802863971181015050224, 7.18787991142440947446010668210, 7.25049481133839713907164713491, 8.032339177978438087498718079872
