| L(s) = 1 | + 2.13·2-s + 2.54·4-s − 3.97·5-s + 1.38·7-s + 1.15·8-s − 8.46·10-s − 1.93·13-s + 2.95·14-s − 2.61·16-s − 1.18·17-s + 2.92·19-s − 10.1·20-s − 7.58·23-s + 10.7·25-s − 4.12·26-s + 3.52·28-s + 8.08·29-s + 9.85·31-s − 7.89·32-s − 2.53·34-s − 5.49·35-s + 5.62·37-s + 6.24·38-s − 4.60·40-s − 12.4·41-s − 0.391·43-s − 16.1·46-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.27·4-s − 1.77·5-s + 0.523·7-s + 0.409·8-s − 2.67·10-s − 0.536·13-s + 0.788·14-s − 0.654·16-s − 0.288·17-s + 0.671·19-s − 2.25·20-s − 1.58·23-s + 2.15·25-s − 0.808·26-s + 0.665·28-s + 1.50·29-s + 1.76·31-s − 1.39·32-s − 0.434·34-s − 0.929·35-s + 0.925·37-s + 1.01·38-s − 0.728·40-s − 1.94·41-s − 0.0596·43-s − 2.38·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.856909655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.856909655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 - 8.08T + 29T^{2} \) |
| 31 | \( 1 - 9.85T + 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 0.391T + 43T^{2} \) |
| 47 | \( 1 + 8.95T + 47T^{2} \) |
| 53 | \( 1 + 3.60T + 53T^{2} \) |
| 59 | \( 1 + 0.103T + 59T^{2} \) |
| 61 | \( 1 - 9.92T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 6.22T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.75609022009966504948251644464, −6.62250803888938466383645843751, −6.49602486848346801316800626788, −5.24615955372548173107500815104, −4.68934219893347057406610335947, −4.38119101687123434357253739674, −3.51691454956066494782729681381, −3.05840665613351973902012868090, −2.07516204955644644071271412911, −0.61562225179393203090572614365,
0.61562225179393203090572614365, 2.07516204955644644071271412911, 3.05840665613351973902012868090, 3.51691454956066494782729681381, 4.38119101687123434357253739674, 4.68934219893347057406610335947, 5.24615955372548173107500815104, 6.49602486848346801316800626788, 6.62250803888938466383645843751, 7.75609022009966504948251644464
