| L(s) = 1 | + 17.1·5-s − 23.1·7-s − 11·11-s − 59.6·13-s − 80.8·17-s − 11.7·19-s − 56.0·23-s + 168.·25-s − 85.4·29-s − 99.8·31-s − 396.·35-s + 402.·37-s + 27.0·41-s − 74·43-s − 408.·47-s + 192.·49-s + 463.·53-s − 188.·55-s − 498.·59-s − 635.·61-s − 1.02e3·65-s − 701.·67-s + 27.7·71-s + 619.·73-s + 254.·77-s − 208.·79-s − 1.30e3·83-s + ⋯ |
| L(s) = 1 | + 1.53·5-s − 1.24·7-s − 0.301·11-s − 1.27·13-s − 1.15·17-s − 0.141·19-s − 0.508·23-s + 1.34·25-s − 0.547·29-s − 0.578·31-s − 1.91·35-s + 1.79·37-s + 0.103·41-s − 0.262·43-s − 1.26·47-s + 0.560·49-s + 1.20·53-s − 0.462·55-s − 1.10·59-s − 1.33·61-s − 1.95·65-s − 1.27·67-s + 0.0463·71-s + 0.993·73-s + 0.376·77-s − 0.296·79-s − 1.72·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 - 17.1T + 125T^{2} \) |
| 7 | \( 1 + 23.1T + 343T^{2} \) |
| 13 | \( 1 + 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 85.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 99.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 27.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 74T + 7.95e4T^{2} \) |
| 47 | \( 1 + 408.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 635.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 701.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 27.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 208.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−10.05783308804332371999961816570, −9.691943699057387933092912451342, −8.904757943762688612686988469143, −7.41064726322153901475281195952, −6.42913890955334638661628131567, −5.76781803290502380869367139818, −4.56896407020150818853070963648, −2.91448017618907003083281129995, −2.01237317689166792041705731153, 0,
2.01237317689166792041705731153, 2.91448017618907003083281129995, 4.56896407020150818853070963648, 5.76781803290502380869367139818, 6.42913890955334638661628131567, 7.41064726322153901475281195952, 8.904757943762688612686988469143, 9.691943699057387933092912451342, 10.05783308804332371999961816570
