| L(s) = 1 | + 0.477·2-s − 1.87·3-s − 1.77·4-s − 5-s − 0.893·6-s − 7-s − 1.80·8-s + 0.508·9-s − 0.477·10-s + 3.31·12-s + 3.98·13-s − 0.477·14-s + 1.87·15-s + 2.68·16-s − 3.06·17-s + 0.242·18-s − 0.112·19-s + 1.77·20-s + 1.87·21-s − 7.20·23-s + 3.37·24-s + 25-s + 1.90·26-s + 4.66·27-s + 1.77·28-s + 3.20·29-s + 0.893·30-s + ⋯ |
| L(s) = 1 | + 0.337·2-s − 1.08·3-s − 0.886·4-s − 0.447·5-s − 0.364·6-s − 0.377·7-s − 0.636·8-s + 0.169·9-s − 0.150·10-s + 0.958·12-s + 1.10·13-s − 0.127·14-s + 0.483·15-s + 0.671·16-s − 0.742·17-s + 0.0571·18-s − 0.0257·19-s + 0.396·20-s + 0.408·21-s − 1.50·23-s + 0.688·24-s + 0.200·25-s + 0.372·26-s + 0.898·27-s + 0.334·28-s + 0.594·29-s + 0.163·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.477T + 2T^{2} \) |
| 3 | \( 1 + 1.87T + 3T^{2} \) |
| 13 | \( 1 - 3.98T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 + 0.112T + 19T^{2} \) |
| 23 | \( 1 + 7.20T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 - 6.55T + 37T^{2} \) |
| 41 | \( 1 + 1.40T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 - 4.59T + 47T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 3.99T + 79T^{2} \) |
| 83 | \( 1 + 3.57T + 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.285252713216961860170682429838, −7.12000830023156076888321469541, −6.24292625645770416612781856493, −5.86372327692341231251748042074, −5.09391202507570489120746783043, −4.18360111156484781917483910267, −3.79566868033481722480736354452, −2.60705228006717337999013772518, −0.970388605069017132984309286139, 0,
0.970388605069017132984309286139, 2.60705228006717337999013772518, 3.79566868033481722480736354452, 4.18360111156484781917483910267, 5.09391202507570489120746783043, 5.86372327692341231251748042074, 6.24292625645770416612781856493, 7.12000830023156076888321469541, 8.285252713216961860170682429838
