| L(s) = 1 | + 2.28·2-s − 2.58·3-s − 2.79·4-s − 5.90·6-s + 27.1·7-s − 24.6·8-s − 20.3·9-s + 11·11-s + 7.22·12-s + 9.09·13-s + 61.9·14-s − 33.8·16-s − 91.1·17-s − 46.3·18-s − 80.9·19-s − 70.2·21-s + 25.0·22-s − 208.·23-s + 63.7·24-s + 20.7·26-s + 122.·27-s − 75.8·28-s + 136.·29-s − 213.·31-s + 119.·32-s − 28.4·33-s − 207.·34-s + ⋯ |
| L(s) = 1 | + 0.806·2-s − 0.497·3-s − 0.349·4-s − 0.401·6-s + 1.46·7-s − 1.08·8-s − 0.752·9-s + 0.301·11-s + 0.173·12-s + 0.194·13-s + 1.18·14-s − 0.528·16-s − 1.30·17-s − 0.606·18-s − 0.977·19-s − 0.729·21-s + 0.243·22-s − 1.88·23-s + 0.541·24-s + 0.156·26-s + 0.872·27-s − 0.511·28-s + 0.872·29-s − 1.23·31-s + 0.661·32-s − 0.150·33-s − 1.04·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 2.28T + 8T^{2} \) |
| 3 | \( 1 + 2.58T + 27T^{2} \) |
| 7 | \( 1 - 27.1T + 343T^{2} \) |
| 13 | \( 1 - 9.09T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 208.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 351.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 1.21T + 6.89e4T^{2} \) |
| 43 | \( 1 + 231.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 283.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 238.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 740.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 446.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 56.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 684.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 428.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 147.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 305.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 205.T + 9.12e5T^{2} \) |
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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
−11.35833461559363289510649610378, −10.28565605243334638181519282200, −8.710979010477534752362927560307, −8.362795277592424680327795198242, −6.65246470465242875657853315457, −5.64443955631968414901410101741, −4.79219684052546232924181651648, −3.88695154171140260814616870886, −2.07944707267507891669780160271, 0,
2.07944707267507891669780160271, 3.88695154171140260814616870886, 4.79219684052546232924181651648, 5.64443955631968414901410101741, 6.65246470465242875657853315457, 8.362795277592424680327795198242, 8.710979010477534752362927560307, 10.28565605243334638181519282200, 11.35833461559363289510649610378
