| L(s) = 1 | − 6.43·3-s − 21.9·5-s + 5.22·7-s + 14.3·9-s + 63.3·11-s − 25.4·13-s + 141.·15-s + 15.9·17-s − 99.7·19-s − 33.6·21-s + 73.4·23-s + 356.·25-s + 81.1·27-s + 159.·29-s + 274.·31-s − 407.·33-s − 114.·35-s − 37·37-s + 163.·39-s − 354.·41-s − 414.·43-s − 315.·45-s + 110.·47-s − 315.·49-s − 102.·51-s − 379.·53-s − 1.39e3·55-s + ⋯ |
| L(s) = 1 | − 1.23·3-s − 1.96·5-s + 0.282·7-s + 0.532·9-s + 1.73·11-s − 0.542·13-s + 2.43·15-s + 0.227·17-s − 1.20·19-s − 0.349·21-s + 0.665·23-s + 2.85·25-s + 0.578·27-s + 1.01·29-s + 1.59·31-s − 2.15·33-s − 0.554·35-s − 0.164·37-s + 0.671·39-s − 1.34·41-s − 1.47·43-s − 1.04·45-s + 0.344·47-s − 0.920·49-s − 0.281·51-s − 0.983·53-s − 3.41·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + 37T \) |
| good | 3 | \( 1 + 6.43T + 27T^{2} \) |
| 5 | \( 1 + 21.9T + 125T^{2} \) |
| 7 | \( 1 - 5.22T + 343T^{2} \) |
| 11 | \( 1 - 63.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 274.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 414.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 379.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 114.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 219.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 628.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 998.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 70.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 407.T + 9.12e5T^{2} \) |
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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
−10.11863386788925621757106674242, −8.708440246044520223214715124177, −8.129367692473973969234455994678, −6.81861617064895480115657089587, −6.54065433874027566195027845835, −4.88630445734674767819110965735, −4.38705447888987805939226478640, −3.28658885884016966106224990084, −1.08188796445974660854911416328, 0,
1.08188796445974660854911416328, 3.28658885884016966106224990084, 4.38705447888987805939226478640, 4.88630445734674767819110965735, 6.54065433874027566195027845835, 6.81861617064895480115657089587, 8.129367692473973969234455994678, 8.708440246044520223214715124177, 10.11863386788925621757106674242
