| L(s) = 1 | + 2.71·2-s − 1.15·3-s + 5.36·4-s − 5-s − 3.14·6-s + 9.14·8-s − 1.65·9-s − 2.71·10-s + 0.381·11-s − 6.21·12-s + 3.35·13-s + 1.15·15-s + 14.0·16-s + 6.22·17-s − 4.49·18-s − 7.85·19-s − 5.36·20-s + 1.03·22-s + 6.01·23-s − 10.5·24-s + 25-s + 9.10·26-s + 5.39·27-s − 4.71·29-s + 3.14·30-s + 4.05·31-s + 19.9·32-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.668·3-s + 2.68·4-s − 0.447·5-s − 1.28·6-s + 3.23·8-s − 0.552·9-s − 0.858·10-s + 0.115·11-s − 1.79·12-s + 0.930·13-s + 0.299·15-s + 3.51·16-s + 1.51·17-s − 1.06·18-s − 1.80·19-s − 1.20·20-s + 0.220·22-s + 1.25·23-s − 2.16·24-s + 0.200·25-s + 1.78·26-s + 1.03·27-s − 0.876·29-s + 0.574·30-s + 0.728·31-s + 3.52·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.022033783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.022033783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 3 | \( 1 + 1.15T + 3T^{2} \) |
| 11 | \( 1 - 0.381T + 11T^{2} \) |
| 13 | \( 1 - 3.35T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 + 7.85T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 - 4.05T + 31T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 - 9.88T + 61T^{2} \) |
| 67 | \( 1 + 5.14T + 67T^{2} \) |
| 71 | \( 1 - 4.99T + 71T^{2} \) |
| 73 | \( 1 + 7.14T + 73T^{2} \) |
| 79 | \( 1 - 0.960T + 79T^{2} \) |
| 83 | \( 1 - 7.91T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.45105347803311957361177333448, −6.58034714743472733274940878947, −6.19825758511312461484884313911, −5.62405877826516063550873254694, −4.93924764155354815333922569527, −4.34999506072022518532633609934, −3.51527333321848802900290432420, −3.07525518415941962542245971863, −2.04835829551865803791858459832, −0.933123817033433928801557907750,
0.933123817033433928801557907750, 2.04835829551865803791858459832, 3.07525518415941962542245971863, 3.51527333321848802900290432420, 4.34999506072022518532633609934, 4.93924764155354815333922569527, 5.62405877826516063550873254694, 6.19825758511312461484884313911, 6.58034714743472733274940878947, 7.45105347803311957361177333448
