| L(s) = 1 | + 9·3-s − 25·5-s + 162.·7-s + 81·9-s − 585.·11-s + 1.04e3·13-s − 225·15-s + 310.·17-s + 239.·19-s + 1.46e3·21-s + 4.80e3·23-s + 625·25-s + 729·27-s − 3.96e3·29-s + 1.15e3·31-s − 5.26e3·33-s − 4.05e3·35-s + 7.16e3·37-s + 9.40e3·39-s + 1.75e4·41-s + 9.78e3·43-s − 2.02e3·45-s + 1.55e4·47-s + 9.55e3·49-s + 2.79e3·51-s − 1.81e4·53-s + 1.46e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.25·7-s + 0.333·9-s − 1.45·11-s + 1.71·13-s − 0.258·15-s + 0.260·17-s + 0.152·19-s + 0.723·21-s + 1.89·23-s + 0.200·25-s + 0.192·27-s − 0.876·29-s + 0.216·31-s − 0.842·33-s − 0.560·35-s + 0.860·37-s + 0.990·39-s + 1.63·41-s + 0.807·43-s − 0.149·45-s + 1.02·47-s + 0.568·49-s + 0.150·51-s − 0.887·53-s + 0.652·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.502821729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502821729\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| good | 7 | \( 1 - 162.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 585.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 310.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 239.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.75e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.78e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.55e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.14e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.11e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.34e4T + 8.58e9T^{2} \) |
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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
−12.75366421285115969011068167676, −11.17462892252557285212086323208, −10.80395342419124902533379327597, −9.099392800209928155301232543114, −8.152480311760291994472945971957, −7.45036133110053448989074028921, −5.61695442556017967373050057504, −4.35186694482245600665525263586, −2.87337773203800534518115890780, −1.18597407581890137401017874695,
1.18597407581890137401017874695, 2.87337773203800534518115890780, 4.35186694482245600665525263586, 5.61695442556017967373050057504, 7.45036133110053448989074028921, 8.152480311760291994472945971957, 9.099392800209928155301232543114, 10.80395342419124902533379327597, 11.17462892252557285212086323208, 12.75366421285115969011068167676
