| L(s) = 1 | + 5.56·2-s + 22.9·4-s + 5·5-s − 7·7-s + 83.0·8-s + 27.8·10-s + 33.6·11-s − 38.3·13-s − 38.9·14-s + 278.·16-s − 65.7·17-s + 33.3·19-s + 114.·20-s + 186.·22-s − 207.·23-s + 25·25-s − 213.·26-s − 160.·28-s + 189.·29-s + 202.·31-s + 883.·32-s − 365.·34-s − 35·35-s − 16.5·37-s + 185.·38-s + 415.·40-s − 388.·41-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.86·4-s + 0.447·5-s − 0.377·7-s + 3.66·8-s + 0.879·10-s + 0.921·11-s − 0.818·13-s − 0.743·14-s + 4.34·16-s − 0.937·17-s + 0.403·19-s + 1.28·20-s + 1.81·22-s − 1.88·23-s + 0.200·25-s − 1.60·26-s − 1.08·28-s + 1.21·29-s + 1.17·31-s + 4.88·32-s − 1.84·34-s − 0.169·35-s − 0.0734·37-s + 0.792·38-s + 1.64·40-s − 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.746874395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.746874395\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 5.56T + 8T^{2} \) |
| 11 | \( 1 - 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 207.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 16.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 388.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 41.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 458.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 256.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 123.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 336.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 22.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 385.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 23.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 51.9T + 9.12e5T^{2} \) |
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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
−11.86058658982462498008428125637, −10.56743904179075894489223368934, −9.710044077043632372652342637899, −8.000205946591197739544235519601, −6.65690065924742606406951462220, −6.31003995061734808623821327526, −5.03730138628468170603228142090, −4.17882472289199866650657227982, −2.98483500504891128244760295134, −1.83275418414447507296403881605,
1.83275418414447507296403881605, 2.98483500504891128244760295134, 4.17882472289199866650657227982, 5.03730138628468170603228142090, 6.31003995061734808623821327526, 6.65690065924742606406951462220, 8.000205946591197739544235519601, 9.710044077043632372652342637899, 10.56743904179075894489223368934, 11.86058658982462498008428125637
