| L(s) = 1 | + 2.53·3-s − 2.27·5-s + 1.27·7-s + 3.42·9-s + 0.683·11-s − 4.05·13-s − 5.77·15-s + 0.471·17-s − 19-s + 3.23·21-s − 4.28·23-s + 0.195·25-s + 1.07·27-s − 9.77·29-s − 9.37·31-s + 1.73·33-s − 2.91·35-s + 2.27·37-s − 10.2·39-s + 1.23·41-s − 0.166·43-s − 7.80·45-s + 5.60·47-s − 5.36·49-s + 1.19·51-s + 53-s − 1.55·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s − 1.01·5-s + 0.482·7-s + 1.14·9-s + 0.206·11-s − 1.12·13-s − 1.49·15-s + 0.114·17-s − 0.229·19-s + 0.706·21-s − 0.893·23-s + 0.0390·25-s + 0.206·27-s − 1.81·29-s − 1.68·31-s + 0.301·33-s − 0.492·35-s + 0.374·37-s − 1.64·39-s + 0.192·41-s − 0.0254·43-s − 1.16·45-s + 0.818·47-s − 0.766·49-s + 0.167·51-s + 0.137·53-s − 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 - 0.683T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 - 0.471T + 17T^{2} \) |
| 23 | \( 1 + 4.28T + 23T^{2} \) |
| 29 | \( 1 + 9.77T + 29T^{2} \) |
| 31 | \( 1 + 9.37T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 0.166T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 59 | \( 1 + 7.90T + 59T^{2} \) |
| 61 | \( 1 - 0.280T + 61T^{2} \) |
| 67 | \( 1 - 2.83T + 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 9.28T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 0.0580T + 97T^{2} \) |
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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
−7.87030544752062737654387038864, −7.69227982511058040211147640405, −7.05163802608605341712699611724, −5.79896088504125264704747409954, −4.84721785457797597362516891570, −3.90297109674562027393205153803, −3.60219916661818220521762568600, −2.44933428493403846138748813447, −1.78360180928001586801246394992, 0,
1.78360180928001586801246394992, 2.44933428493403846138748813447, 3.60219916661818220521762568600, 3.90297109674562027393205153803, 4.84721785457797597362516891570, 5.79896088504125264704747409954, 7.05163802608605341712699611724, 7.69227982511058040211147640405, 7.87030544752062737654387038864
