| L(s) = 1 | − 2.39·2-s + 3.72·4-s − 4.15·7-s − 4.13·8-s + 5.71·11-s − 3.79·13-s + 9.93·14-s + 2.44·16-s − 2.66·17-s + 19-s − 13.6·22-s + 8.13·23-s + 9.09·26-s − 15.4·28-s − 4.73·29-s − 2.31·31-s + 2.42·32-s + 6.38·34-s + 4.68·37-s − 2.39·38-s − 10.1·41-s + 1.76·43-s + 21.3·44-s − 19.4·46-s + 7.14·47-s + 10.2·49-s − 14.1·52-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.86·4-s − 1.56·7-s − 1.46·8-s + 1.72·11-s − 1.05·13-s + 2.65·14-s + 0.610·16-s − 0.646·17-s + 0.229·19-s − 2.91·22-s + 1.69·23-s + 1.78·26-s − 2.92·28-s − 0.878·29-s − 0.415·31-s + 0.428·32-s + 1.09·34-s + 0.770·37-s − 0.388·38-s − 1.59·41-s + 0.268·43-s + 3.21·44-s − 2.87·46-s + 1.04·47-s + 1.46·49-s − 1.96·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5189929554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5189929554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 23 | \( 1 - 8.13T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 - 0.582T + 59T^{2} \) |
| 61 | \( 1 + 9.54T + 61T^{2} \) |
| 67 | \( 1 - 1.20T + 67T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 5.06T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 + 3.36T + 89T^{2} \) |
| 97 | \( 1 + 0.313T + 97T^{2} \) |
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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
−8.794625537918236448248130378105, −7.60633459694852004844426796794, −6.95474732837638428356189960021, −6.69067481538171091121650026524, −5.85173804224016612991949361319, −4.56620065331886302423011987095, −3.49487377140291787491351725876, −2.70576963240504502999320286165, −1.62305936341702906789237253445, −0.52954054366891102187890226276,
0.52954054366891102187890226276, 1.62305936341702906789237253445, 2.70576963240504502999320286165, 3.49487377140291787491351725876, 4.56620065331886302423011987095, 5.85173804224016612991949361319, 6.69067481538171091121650026524, 6.95474732837638428356189960021, 7.60633459694852004844426796794, 8.794625537918236448248130378105
