| L(s) = 1 | + 16·4-s + 41·5-s − 49·7-s + 81·9-s − 169·13-s + 256·16-s + 97·19-s + 656·20-s + 967·23-s + 1.05e3·25-s − 784·28-s − 593·29-s − 1.10e3·31-s − 2.00e3·35-s + 1.29e3·36-s + 2.46e3·41-s − 3.67e3·43-s + 3.32e3·45-s − 2.14e3·47-s + 2.40e3·49-s − 2.70e3·52-s − 5.39e3·53-s − 1.13e3·59-s − 3.96e3·63-s + 4.09e3·64-s − 6.92e3·65-s + 9.81e3·73-s + ⋯ |
| L(s) = 1 | + 4-s + 1.63·5-s − 7-s + 9-s − 13-s + 16-s + 0.268·19-s + 1.63·20-s + 1.82·23-s + 1.68·25-s − 28-s − 0.705·29-s − 1.14·31-s − 1.63·35-s + 36-s + 1.46·41-s − 1.98·43-s + 1.63·45-s − 0.970·47-s + 49-s − 52-s − 1.91·53-s − 0.326·59-s − 63-s + 64-s − 1.63·65-s + 1.84·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.568307757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.568307757\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 5 | \( 1 - 41 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 - 97 T + p^{4} T^{2} \) |
| 23 | \( 1 - 967 T + p^{4} T^{2} \) |
| 29 | \( 1 + 593 T + p^{4} T^{2} \) |
| 31 | \( 1 + 1103 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( 1 - 2462 T + p^{4} T^{2} \) |
| 43 | \( 1 + 3673 T + p^{4} T^{2} \) |
| 47 | \( 1 + 2143 T + p^{4} T^{2} \) |
| 53 | \( 1 + 5393 T + p^{4} T^{2} \) |
| 59 | \( 1 + 1138 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 9817 T + p^{4} T^{2} \) |
| 79 | \( 1 + 7993 T + p^{4} T^{2} \) |
| 83 | \( 1 + 11503 T + p^{4} T^{2} \) |
| 89 | \( 1 + 11383 T + p^{4} T^{2} \) |
| 97 | \( 1 - 1657 T + p^{4} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.04643174493185848731001845400, −12.67696594162749887034309699856, −11.01195011315370979960498541434, −9.927962684909888507917798209587, −9.417543296442854553616825405045, −7.24484458923481231762787443680, −6.50596123998285886600341052584, −5.28693320782193333808137338191, −2.96010362954622717678977482941, −1.64269271324134149011256450894,
1.64269271324134149011256450894, 2.96010362954622717678977482941, 5.28693320782193333808137338191, 6.50596123998285886600341052584, 7.24484458923481231762787443680, 9.417543296442854553616825405045, 9.927962684909888507917798209587, 11.01195011315370979960498541434, 12.67696594162749887034309699856, 13.04643174493185848731001845400
