| L(s) = 1 | + 2-s − 3.25·3-s + 4-s + 3.43·5-s − 3.25·6-s + 2.34·7-s + 8-s + 7.59·9-s + 3.43·10-s + 4.43·11-s − 3.25·12-s − 7.18·13-s + 2.34·14-s − 11.1·15-s + 16-s − 5.19·17-s + 7.59·18-s + 2.28·19-s + 3.43·20-s − 7.62·21-s + 4.43·22-s + 23-s − 3.25·24-s + 6.76·25-s − 7.18·26-s − 14.9·27-s + 2.34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.87·3-s + 0.5·4-s + 1.53·5-s − 1.32·6-s + 0.885·7-s + 0.353·8-s + 2.53·9-s + 1.08·10-s + 1.33·11-s − 0.939·12-s − 1.99·13-s + 0.626·14-s − 2.88·15-s + 0.250·16-s − 1.26·17-s + 1.78·18-s + 0.524·19-s + 0.767·20-s − 1.66·21-s + 0.944·22-s + 0.208·23-s − 0.664·24-s + 1.35·25-s − 1.40·26-s − 2.87·27-s + 0.442·28-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
| \(L(1)\) |
\(\approx\) |
\(2.697961068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.697961068\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 + 7.18T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 29 | \( 1 - 6.34T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 4.13T + 37T^{2} \) |
| 41 | \( 1 - 5.21T + 41T^{2} \) |
| 43 | \( 1 - 4.75T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 79 | \( 1 + 8.12T + 79T^{2} \) |
| 83 | \( 1 + 0.119T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−7.61121758839455182756474633099, −6.97029629644636923332860117407, −6.45685476811034084517028057922, −5.84686029941240993248280958222, −5.22689220241555743693405249745, −4.69223759680451797022221621980, −4.22130176280436204875772715890, −2.49551087241159219399970813774, −1.80146243079840691719154427392, −0.897740946834297631998509707429,
0.897740946834297631998509707429, 1.80146243079840691719154427392, 2.49551087241159219399970813774, 4.22130176280436204875772715890, 4.69223759680451797022221621980, 5.22689220241555743693405249745, 5.84686029941240993248280958222, 6.45685476811034084517028057922, 6.97029629644636923332860117407, 7.61121758839455182756474633099
