| L(s) = 1 | + 2·3-s + 50·5-s − 98·7-s − 47·9-s − 254·11-s − 1.34e3·13-s + 100·15-s − 1.78e3·17-s + 1.97e3·19-s − 196·21-s − 2.11e3·23-s + 1.87e3·25-s + 290·27-s − 4.76e3·29-s + 1.36e4·31-s − 508·33-s − 4.90e3·35-s + 2.13e3·37-s − 2.68e3·39-s − 6.74e3·41-s − 2.67e4·43-s − 2.35e3·45-s − 6.32e3·47-s + 7.20e3·49-s − 3.57e3·51-s + 2.46e4·53-s − 1.27e4·55-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 0.894·5-s − 0.755·7-s − 0.193·9-s − 0.632·11-s − 2.20·13-s + 0.114·15-s − 1.49·17-s + 1.25·19-s − 0.0969·21-s − 0.832·23-s + 3/5·25-s + 0.0765·27-s − 1.05·29-s + 2.55·31-s − 0.0812·33-s − 0.676·35-s + 0.256·37-s − 0.282·39-s − 0.626·41-s − 2.20·43-s − 0.172·45-s − 0.417·47-s + 3/7·49-s − 0.192·51-s + 1.20·53-s − 0.566·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 17 p T^{2} - 2 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 254 T + 196967 T^{2} + 254 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1342 T + 1157511 T^{2} + 1342 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1786 T + 3270487 T^{2} + 1786 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 104 p T + 5394242 T^{2} - 104 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2112 T + 5319706 T^{2} + 2112 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4762 T + 43001155 T^{2} + 4762 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 13692 T + 103651214 T^{2} - 13692 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2136 T + 76538342 T^{2} - 2136 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6740 T + 226828738 T^{2} + 6740 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 26728 T + 10895606 p T^{2} + 26728 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6326 T + 341085667 T^{2} + 6326 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 24624 T + 804898186 T^{2} - 24624 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 51336 T + 1514088678 T^{2} - 51336 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4468 T + 1171408458 T^{2} - 4468 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 39168 T + 2464286566 T^{2} + 39168 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 41232 T + 1299751182 T^{2} - 41232 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 36124 T + 2288610246 T^{2} + 36124 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 140842 T + 11056094855 T^{2} + 140842 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 57712 T + 6487776118 T^{2} + 57712 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20236 T + 11130323618 T^{2} + 20236 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 183586 T + 21808917327 T^{2} + 183586 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.53716480563578443316764959167, −9.989926544160370132370361190853, −9.812408711866967395363809002887, −9.773044483547376559140304989930, −8.795047120568212939547029032805, −8.525062050498309552010755000038, −7.83880408036866029092187279991, −7.23742630211986031196157855507, −6.78911766124088064142510895916, −6.46102879792408123407244983989, −5.50623254997024427727622418445, −5.37378632764922417859751594880, −4.64768749318429360500441583155, −4.10714946242204420527586907919, −3.00997703368923311194488693382, −2.66489186607734107654184784898, −2.20748459355736745805899095861, −1.29847209752911182738128036415, 0, 0,
1.29847209752911182738128036415, 2.20748459355736745805899095861, 2.66489186607734107654184784898, 3.00997703368923311194488693382, 4.10714946242204420527586907919, 4.64768749318429360500441583155, 5.37378632764922417859751594880, 5.50623254997024427727622418445, 6.46102879792408123407244983989, 6.78911766124088064142510895916, 7.23742630211986031196157855507, 7.83880408036866029092187279991, 8.525062050498309552010755000038, 8.795047120568212939547029032805, 9.773044483547376559140304989930, 9.812408711866967395363809002887, 9.989926544160370132370361190853, 10.53716480563578443316764959167
