| L(s) = 1 | + 2.54·3-s + 3.47·9-s − 3.91·11-s − 3.47·13-s + 5.08·17-s + 5.25·19-s − 6.01·23-s + 1.20·27-s − 10.2·29-s − 2.04·31-s − 9.95·33-s + 0.405·37-s − 8.83·39-s + 2.57·41-s − 6.15·43-s + 0.480·47-s + 12.9·51-s + 0.466·53-s + 13.3·57-s + 10.3·59-s − 8.08·61-s + 4.15·67-s − 15.3·69-s − 4.28·71-s − 4.81·73-s − 12.6·79-s − 7.35·81-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.15·9-s − 1.17·11-s − 0.963·13-s + 1.23·17-s + 1.20·19-s − 1.25·23-s + 0.231·27-s − 1.90·29-s − 0.367·31-s − 1.73·33-s + 0.0665·37-s − 1.41·39-s + 0.402·41-s − 0.939·43-s + 0.0700·47-s + 1.81·51-s + 0.0640·53-s + 1.76·57-s + 1.34·59-s − 1.03·61-s + 0.507·67-s − 1.84·69-s − 0.508·71-s − 0.563·73-s − 1.42·79-s − 0.817·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 23 | \( 1 + 6.01T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 - 0.405T + 37T^{2} \) |
| 41 | \( 1 - 2.57T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 - 0.480T + 47T^{2} \) |
| 53 | \( 1 - 0.466T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 8.08T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 + 4.28T + 71T^{2} \) |
| 73 | \( 1 + 4.81T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 + 1.32T + 97T^{2} \) |
| 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
−7.53014423165467032444634425774, −7.09277182575219883829880654691, −5.63478120807311374323172150876, −5.47380443858001091952338009154, −4.38017374670617440157245970540, −3.57268334038515846670135629048, −3.00557726906082864455579955483, −2.32676912548192635856048572164, −1.54447315937560429442657434389, 0,
1.54447315937560429442657434389, 2.32676912548192635856048572164, 3.00557726906082864455579955483, 3.57268334038515846670135629048, 4.38017374670617440157245970540, 5.47380443858001091952338009154, 5.63478120807311374323172150876, 7.09277182575219883829880654691, 7.53014423165467032444634425774
