| L(s) = 1 | + 2.49·2-s − 0.300·3-s + 4.21·4-s − 0.749·6-s − 2.12·7-s + 5.50·8-s − 2.90·9-s − 4.14·11-s − 1.26·12-s + 6.32·13-s − 5.28·14-s + 5.30·16-s + 6.52·17-s − 7.25·18-s − 7.41·19-s + 0.638·21-s − 10.3·22-s − 8.62·23-s − 1.65·24-s + 15.7·26-s + 1.77·27-s − 8.93·28-s + 2.32·29-s − 8.18·31-s + 2.21·32-s + 1.24·33-s + 16.2·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.173·3-s + 2.10·4-s − 0.305·6-s − 0.802·7-s + 1.94·8-s − 0.969·9-s − 1.24·11-s − 0.365·12-s + 1.75·13-s − 1.41·14-s + 1.32·16-s + 1.58·17-s − 1.70·18-s − 1.70·19-s + 0.139·21-s − 2.20·22-s − 1.79·23-s − 0.338·24-s + 3.08·26-s + 0.341·27-s − 1.68·28-s + 0.431·29-s − 1.46·31-s + 0.391·32-s + 0.216·33-s + 2.79·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 0.300T + 3T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 6.32T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 + 7.41T + 19T^{2} \) |
| 23 | \( 1 + 8.62T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 + 8.18T + 31T^{2} \) |
| 37 | \( 1 + 0.305T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 8.13T + 43T^{2} \) |
| 47 | \( 1 + 1.97T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 6.96T + 59T^{2} \) |
| 61 | \( 1 + 6.81T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 8.36T + 71T^{2} \) |
| 73 | \( 1 + 4.06T + 73T^{2} \) |
| 79 | \( 1 - 5.10T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 + 5.08T + 97T^{2} \) |
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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
−7.60807740037505082686775421958, −6.46636742222327976547282207069, −6.05559063270029137146487736868, −5.69190401845294761568503807744, −4.92108406549975352544245085204, −3.85865044504209451667396503074, −3.47356295012780854691924027843, −2.71974575492140031878342634196, −1.79780519901831242090408830839, 0,
1.79780519901831242090408830839, 2.71974575492140031878342634196, 3.47356295012780854691924027843, 3.85865044504209451667396503074, 4.92108406549975352544245085204, 5.69190401845294761568503807744, 6.05559063270029137146487736868, 6.46636742222327976547282207069, 7.60807740037505082686775421958
