| L(s) = 1 | + 1.25·2-s + 2.42·3-s − 0.418·4-s + 3.05·6-s − 3.44·7-s − 3.04·8-s + 2.90·9-s + 6.10·11-s − 1.01·12-s − 2.91·13-s − 4.32·14-s − 2.98·16-s − 2.89·17-s + 3.65·18-s + 0.435·19-s − 8.35·21-s + 7.67·22-s + 0.0375·23-s − 7.39·24-s − 3.66·26-s − 0.232·27-s + 1.44·28-s − 1.49·29-s − 3.02·31-s + 2.32·32-s + 14.8·33-s − 3.63·34-s + ⋯ |
| L(s) = 1 | + 0.889·2-s + 1.40·3-s − 0.209·4-s + 1.24·6-s − 1.30·7-s − 1.07·8-s + 0.968·9-s + 1.84·11-s − 0.293·12-s − 0.807·13-s − 1.15·14-s − 0.746·16-s − 0.701·17-s + 0.860·18-s + 0.0998·19-s − 1.82·21-s + 1.63·22-s + 0.00782·23-s − 1.50·24-s − 0.718·26-s − 0.0446·27-s + 0.272·28-s − 0.278·29-s − 0.543·31-s + 0.411·32-s + 2.58·33-s − 0.623·34-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 | \( 1 - 1.25T + 2T^{2} \) |
| 3 | \( 1 - 2.42T + 3T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 0.435T + 19T^{2} \) |
| 23 | \( 1 - 0.0375T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 31 | \( 1 + 3.02T + 31T^{2} \) |
| 37 | \( 1 + 6.11T + 37T^{2} \) |
| 41 | \( 1 + 0.436T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 - 0.115T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 - 0.262T + 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 + 3.99T + 83T^{2} \) |
| 89 | \( 1 + 8.60T + 89T^{2} \) |
| 97 | \( 1 + 0.615T + 97T^{2} \) |
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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
−7.68624904937993900214507071802, −6.82002248083705415277105023207, −6.43492221829117556982548232188, −5.53424565401891541774027664627, −4.47651944565020236048183522322, −3.90381106099559273052289052401, −3.32759604665877476795895765680, −2.75833170147421174413230863464, −1.69538514571676628647938085930, 0,
1.69538514571676628647938085930, 2.75833170147421174413230863464, 3.32759604665877476795895765680, 3.90381106099559273052289052401, 4.47651944565020236048183522322, 5.53424565401891541774027664627, 6.43492221829117556982548232188, 6.82002248083705415277105023207, 7.68624904937993900214507071802
