| L(s) = 1 | − 1.91·2-s − 2.10·3-s + 1.65·4-s + 4.02·6-s − 2.16·7-s + 0.668·8-s + 1.42·9-s − 4.26·11-s − 3.47·12-s − 4.06·13-s + 4.13·14-s − 4.57·16-s − 5.01·17-s − 2.72·18-s − 0.180·19-s + 4.55·21-s + 8.15·22-s − 2.32·23-s − 1.40·24-s + 7.76·26-s + 3.30·27-s − 3.57·28-s − 0.555·29-s + 3.29·31-s + 7.40·32-s + 8.98·33-s + 9.58·34-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 1.21·3-s + 0.825·4-s + 1.64·6-s − 0.818·7-s + 0.236·8-s + 0.476·9-s − 1.28·11-s − 1.00·12-s − 1.12·13-s + 1.10·14-s − 1.14·16-s − 1.21·17-s − 0.643·18-s − 0.0414·19-s + 0.993·21-s + 1.73·22-s − 0.484·23-s − 0.286·24-s + 1.52·26-s + 0.636·27-s − 0.675·28-s − 0.103·29-s + 0.591·31-s + 1.30·32-s + 1.56·33-s + 1.64·34-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 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 + 2.10T + 3T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 + 4.06T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 19 | \( 1 + 0.180T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 + 0.555T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 - 3.48T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 - 4.61T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 - 1.18T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 8.04T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82041010995615950970672007402, −7.03684505188055221043407465990, −6.49349179904781197408858130624, −5.74643904752627873964918588212, −4.90209551391339263496356817968, −4.33837161362173336119659144714, −2.81963684033002402294075245129, −2.15818201381579757498240179760, −0.66350276924624769237570721951, 0,
0.66350276924624769237570721951, 2.15818201381579757498240179760, 2.81963684033002402294075245129, 4.33837161362173336119659144714, 4.90209551391339263496356817968, 5.74643904752627873964918588212, 6.49349179904781197408858130624, 7.03684505188055221043407465990, 7.82041010995615950970672007402
