| L(s) = 1 | + 1.85·2-s − 4.55·4-s + 1.02·7-s − 23.3·8-s − 16.1·11-s + 3.05·13-s + 1.90·14-s − 6.86·16-s + 69.2·17-s − 12.6·19-s − 29.9·22-s − 56.6·23-s + 5.67·26-s − 4.66·28-s + 196.·29-s + 213.·31-s + 173.·32-s + 128.·34-s + 299.·37-s − 23.4·38-s − 139.·41-s − 475.·43-s + 73.4·44-s − 105.·46-s + 193.·47-s − 341.·49-s − 13.9·52-s + ⋯ |
| L(s) = 1 | + 0.656·2-s − 0.568·4-s + 0.0552·7-s − 1.03·8-s − 0.442·11-s + 0.0651·13-s + 0.0363·14-s − 0.107·16-s + 0.988·17-s − 0.152·19-s − 0.290·22-s − 0.513·23-s + 0.0427·26-s − 0.0314·28-s + 1.25·29-s + 1.23·31-s + 0.959·32-s + 0.648·34-s + 1.33·37-s − 0.100·38-s − 0.529·41-s − 1.68·43-s + 0.251·44-s − 0.337·46-s + 0.600·47-s − 0.996·49-s − 0.0370·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.85T + 8T^{2} \) |
| 7 | \( 1 - 1.02T + 343T^{2} \) |
| 11 | \( 1 + 16.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.05T + 2.19e3T^{2} \) |
| 17 | \( 1 - 69.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 475.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 214.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 495.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 761.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 736.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 701.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 780.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 961.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 520.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.255586773094212395994620149573, −7.84406132213336205680060971733, −6.53915480791500276140913216383, −5.94713014253938596916890194680, −4.99682848607169956750481377008, −4.46890982389855397666011005051, −3.41482981098137559166162063724, −2.70140169839067378693604744769, −1.20919751871574537126153155428, 0,
1.20919751871574537126153155428, 2.70140169839067378693604744769, 3.41482981098137559166162063724, 4.46890982389855397666011005051, 4.99682848607169956750481377008, 5.94713014253938596916890194680, 6.53915480791500276140913216383, 7.84406132213336205680060971733, 8.255586773094212395994620149573
