| L(s) = 1 | + 12·9-s − 8·11-s − 12·25-s + 40·43-s − 12·49-s + 24·67-s + 64·81-s − 96·99-s − 56·107-s + 72·113-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 92·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 4·9-s − 2.41·11-s − 2.39·25-s + 6.09·43-s − 1.71·49-s + 2.93·67-s + 64/9·81-s − 9.64·99-s − 5.41·107-s + 6.77·113-s − 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 7.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.659151454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659151454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 2 p T^{2} + 22 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 6 T^{2} + 14 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 44 T^{2} + 982 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 46 T^{2} + 1126 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 44 T^{2} + 1846 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 4 T^{2} + 3702 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 110 T^{2} + 6342 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 10 T^{2} + 5262 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 148 T^{2} + 12838 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 102 T^{2} + 15254 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 20 T^{2} + 10822 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 364 T^{2} + 51862 T^{4} - 364 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.90965075793879032396067784441, −3.78340719133862026100434034225, −3.72693341975072553891185994495, −3.66805248865687938681057844796, −3.47330818882741549023142934843, −3.38172796193466465955839180716, −3.28664061341233643649506078162, −2.84610049585545026101164252010, −2.78160224726447443572080566753, −2.66964876731016024832518495322, −2.65502475108992766539414905594, −2.50440409343932148086244856960, −2.17975388282180494943632996316, −2.16750271224823091257697284772, −2.16742973309704941628265352859, −2.08164633212750294565993622419, −1.80332418455986277362875748938, −1.43833525477864567947238136382, −1.33348151928695220980518276838, −1.26061396776634108531943647913, −1.24592202681324641511054577890, −0.956338335179852325575628291956, −0.72229309382812916577396739738, −0.40629533843496138244258117427, −0.15053470445527033493281457802,
0.15053470445527033493281457802, 0.40629533843496138244258117427, 0.72229309382812916577396739738, 0.956338335179852325575628291956, 1.24592202681324641511054577890, 1.26061396776634108531943647913, 1.33348151928695220980518276838, 1.43833525477864567947238136382, 1.80332418455986277362875748938, 2.08164633212750294565993622419, 2.16742973309704941628265352859, 2.16750271224823091257697284772, 2.17975388282180494943632996316, 2.50440409343932148086244856960, 2.65502475108992766539414905594, 2.66964876731016024832518495322, 2.78160224726447443572080566753, 2.84610049585545026101164252010, 3.28664061341233643649506078162, 3.38172796193466465955839180716, 3.47330818882741549023142934843, 3.66805248865687938681057844796, 3.72693341975072553891185994495, 3.78340719133862026100434034225, 3.90965075793879032396067784441
Plot not available for L-functions of degree greater than 10.
