| L(s) = 1 | + 0.476·2-s + 1.43·3-s − 1.77·4-s + 5-s + 0.683·6-s + 4.32·7-s − 1.79·8-s − 0.943·9-s + 0.476·10-s + 1.24·11-s − 2.54·12-s − 2.16·13-s + 2.05·14-s + 1.43·15-s + 2.68·16-s + 4.28·17-s − 0.449·18-s + 2.45·19-s − 1.77·20-s + 6.19·21-s + 0.592·22-s + 6.16·23-s − 2.57·24-s + 25-s − 1.03·26-s − 5.65·27-s − 7.66·28-s + ⋯ |
| L(s) = 1 | + 0.336·2-s + 0.827·3-s − 0.886·4-s + 0.447·5-s + 0.278·6-s + 1.63·7-s − 0.635·8-s − 0.314·9-s + 0.150·10-s + 0.375·11-s − 0.733·12-s − 0.599·13-s + 0.550·14-s + 0.370·15-s + 0.672·16-s + 1.04·17-s − 0.105·18-s + 0.563·19-s − 0.396·20-s + 1.35·21-s + 0.126·22-s + 1.28·23-s − 0.526·24-s + 0.200·25-s − 0.202·26-s − 1.08·27-s − 1.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.381855717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.381855717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 - 0.476T + 2T^{2} \) |
| 3 | \( 1 - 1.43T + 3T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 + 6.50T + 37T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.592T + 71T^{2} \) |
| 73 | \( 1 + 6.68T + 73T^{2} \) |
| 79 | \( 1 + 3.56T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−10.08387954474786248385912357143, −9.422329190039158636177563605875, −8.449845941818269157485180876433, −8.168947146215667007638833833879, −7.00273414387778769022804493298, −5.35200545490203191191854964337, −5.13793697911971185578677828612, −3.86322928784422490016143672383, −2.80871338647371157878412442674, −1.39789695314488501104990126287,
1.39789695314488501104990126287, 2.80871338647371157878412442674, 3.86322928784422490016143672383, 5.13793697911971185578677828612, 5.35200545490203191191854964337, 7.00273414387778769022804493298, 8.168947146215667007638833833879, 8.449845941818269157485180876433, 9.422329190039158636177563605875, 10.08387954474786248385912357143
