| L(s) = 1 | + 2.46·2-s − 3-s + 4.09·4-s + 5-s − 2.46·6-s + 2.10·7-s + 5.17·8-s + 9-s + 2.46·10-s + 2.71·11-s − 4.09·12-s − 2.86·13-s + 5.18·14-s − 15-s + 4.59·16-s − 4.29·17-s + 2.46·18-s + 2.48·19-s + 4.09·20-s − 2.10·21-s + 6.70·22-s + 3.62·23-s − 5.17·24-s + 25-s − 7.08·26-s − 27-s + 8.61·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.04·4-s + 0.447·5-s − 1.00·6-s + 0.794·7-s + 1.83·8-s + 0.333·9-s + 0.780·10-s + 0.818·11-s − 1.18·12-s − 0.795·13-s + 1.38·14-s − 0.258·15-s + 1.14·16-s − 1.04·17-s + 0.582·18-s + 0.569·19-s + 0.916·20-s − 0.458·21-s + 1.42·22-s + 0.756·23-s − 1.05·24-s + 0.200·25-s − 1.38·26-s − 0.192·27-s + 1.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.363058434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.363058434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 - 4.66T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 43 | \( 1 + 1.05T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 1.24T + 61T^{2} \) |
| 67 | \( 1 + 1.33T + 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 - 0.809T + 73T^{2} \) |
| 79 | \( 1 + 1.45T + 79T^{2} \) |
| 83 | \( 1 - 2.87T + 83T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 + 8.40T + 97T^{2} \) |
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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
−7.72830014689028497068456855128, −6.93625318176085961720172780623, −6.48126932118831907437977726987, −5.80661662283722122209855790635, −4.96121397854620306712396823978, −4.69055609580629348654820242042, −3.95688453621417520132005566519, −2.87124383837096726473707108041, −2.16326116166868100169379014670, −1.12530509815724655663864619529,
1.12530509815724655663864619529, 2.16326116166868100169379014670, 2.87124383837096726473707108041, 3.95688453621417520132005566519, 4.69055609580629348654820242042, 4.96121397854620306712396823978, 5.80661662283722122209855790635, 6.48126932118831907437977726987, 6.93625318176085961720172780623, 7.72830014689028497068456855128
