| L(s) = 1 | − 2-s − 0.554·3-s + 4-s + 0.198·5-s + 0.554·6-s + 0.109·7-s − 8-s − 2.69·9-s − 0.198·10-s − 1.33·11-s − 0.554·12-s + 3.93·13-s − 0.109·14-s − 0.109·15-s + 16-s − 2.91·17-s + 2.69·18-s + 1.29·19-s + 0.198·20-s − 0.0609·21-s + 1.33·22-s + 7.78·23-s + 0.554·24-s − 4.96·25-s − 3.93·26-s + 3.15·27-s + 0.109·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.320·3-s + 0.5·4-s + 0.0885·5-s + 0.226·6-s + 0.0415·7-s − 0.353·8-s − 0.897·9-s − 0.0626·10-s − 0.402·11-s − 0.160·12-s + 1.09·13-s − 0.0293·14-s − 0.0283·15-s + 0.250·16-s − 0.706·17-s + 0.634·18-s + 0.297·19-s + 0.0442·20-s − 0.0133·21-s + 0.284·22-s + 1.62·23-s + 0.113·24-s − 0.992·25-s − 0.772·26-s + 0.607·27-s + 0.0207·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 0.554T + 3T^{2} \) |
| 5 | \( 1 - 0.198T + 5T^{2} \) |
| 7 | \( 1 - 0.109T + 7T^{2} \) |
| 11 | \( 1 + 1.33T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 + 3.02T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 - 0.801T + 47T^{2} \) |
| 53 | \( 1 - 8.33T + 53T^{2} \) |
| 59 | \( 1 + 5.08T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.94T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.929240226752559219216606985179, −8.335465409885209652244035419360, −7.43148732303967491581870668300, −6.58618021968648013526672222445, −5.78655308689025560120034542014, −5.06880998106689682645097849347, −3.69019614019029915776147969721, −2.73892563133134694981633355845, −1.48630710283875076660165321951, 0,
1.48630710283875076660165321951, 2.73892563133134694981633355845, 3.69019614019029915776147969721, 5.06880998106689682645097849347, 5.78655308689025560120034542014, 6.58618021968648013526672222445, 7.43148732303967491581870668300, 8.335465409885209652244035419360, 8.929240226752559219216606985179
