| L(s) = 1 | + 5.46·3-s − 12.9·5-s − 28.3·7-s + 2.85·9-s − 55.0·11-s − 0.430·13-s − 70.6·15-s + 17·17-s + 147.·19-s − 155.·21-s + 108.·23-s + 42.1·25-s − 131.·27-s − 107.·29-s − 70.1·31-s − 300.·33-s + 367.·35-s − 381.·37-s − 2.35·39-s − 16.1·41-s + 382.·43-s − 36.9·45-s − 455.·47-s + 463.·49-s + 92.8·51-s + 21.0·53-s + 711.·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s − 1.15·5-s − 1.53·7-s + 0.105·9-s − 1.50·11-s − 0.00919·13-s − 1.21·15-s + 0.242·17-s + 1.78·19-s − 1.61·21-s + 0.984·23-s + 0.337·25-s − 0.940·27-s − 0.691·29-s − 0.406·31-s − 1.58·33-s + 1.77·35-s − 1.69·37-s − 0.00966·39-s − 0.0614·41-s + 1.35·43-s − 0.122·45-s − 1.41·47-s + 1.35·49-s + 0.255·51-s + 0.0544·53-s + 1.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 5.46T + 27T^{2} \) |
| 5 | \( 1 + 12.9T + 125T^{2} \) |
| 7 | \( 1 + 28.3T + 343T^{2} \) |
| 11 | \( 1 + 55.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.430T + 2.19e3T^{2} \) |
| 19 | \( 1 - 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 70.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 16.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 382.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 21.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 9.91T + 2.05e5T^{2} \) |
| 61 | \( 1 + 679.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 708.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 85.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 37.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 685.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 175.T + 9.12e5T^{2} \) |
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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
−12.46764324905871366406643248770, −11.23176205927366584067981330293, −9.953914877173093166775737652918, −9.046126559001149467984437485107, −7.87932884677054957333475707791, −7.19694506873248179453620676188, −5.41401985460512562094356725340, −3.52491421178731039093130966559, −2.93688891960522502344082134268, 0,
2.93688891960522502344082134268, 3.52491421178731039093130966559, 5.41401985460512562094356725340, 7.19694506873248179453620676188, 7.87932884677054957333475707791, 9.046126559001149467984437485107, 9.953914877173093166775737652918, 11.23176205927366584067981330293, 12.46764324905871366406643248770
