| L(s) = 1 | + 2-s − 0.686·3-s + 4-s + 1.74·5-s − 0.686·6-s − 1.83·7-s + 8-s − 2.52·9-s + 1.74·10-s − 0.0611·11-s − 0.686·12-s − 3.93·13-s − 1.83·14-s − 1.19·15-s + 16-s − 0.265·17-s − 2.52·18-s + 3.72·19-s + 1.74·20-s + 1.25·21-s − 0.0611·22-s + 23-s − 0.686·24-s − 1.96·25-s − 3.93·26-s + 3.79·27-s − 1.83·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.396·3-s + 0.5·4-s + 0.779·5-s − 0.280·6-s − 0.693·7-s + 0.353·8-s − 0.842·9-s + 0.551·10-s − 0.0184·11-s − 0.198·12-s − 1.09·13-s − 0.490·14-s − 0.309·15-s + 0.250·16-s − 0.0644·17-s − 0.596·18-s + 0.854·19-s + 0.389·20-s + 0.274·21-s − 0.0130·22-s + 0.208·23-s − 0.140·24-s − 0.392·25-s − 0.772·26-s + 0.730·27-s − 0.346·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.686T + 3T^{2} \) |
| 5 | \( 1 - 1.74T + 5T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + 0.0611T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 + 0.265T + 17T^{2} \) |
| 19 | \( 1 - 3.72T + 19T^{2} \) |
| 29 | \( 1 - 6.96T + 29T^{2} \) |
| 31 | \( 1 + 0.157T + 31T^{2} \) |
| 37 | \( 1 - 5.14T + 37T^{2} \) |
| 41 | \( 1 - 0.251T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 0.940T + 53T^{2} \) |
| 59 | \( 1 + 0.819T + 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 + 0.387T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 - 1.13T + 79T^{2} \) |
| 83 | \( 1 + 9.25T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 9.95T + 97T^{2} \) |
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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
−7.57012117074511813549172224080, −6.67506392556273103519705614568, −6.28884034856406851263544602863, −5.46743319879121051432565545029, −5.07725556087062905473434840102, −4.15576434656803956874342905319, −2.93664276198418568025685065001, −2.73026466279800405208312450787, −1.45535210193165130837851126370, 0,
1.45535210193165130837851126370, 2.73026466279800405208312450787, 2.93664276198418568025685065001, 4.15576434656803956874342905319, 5.07725556087062905473434840102, 5.46743319879121051432565545029, 6.28884034856406851263544602863, 6.67506392556273103519705614568, 7.57012117074511813549172224080
