| L(s) = 1 | + 10.2·2-s + 73.3·4-s − 68.9·7-s + 423.·8-s + 486.·11-s + 428.·13-s − 707.·14-s + 2.00e3·16-s + 1.80e3·17-s − 1.04e3·19-s + 4.98e3·22-s + 686.·23-s + 4.39e3·26-s − 5.05e3·28-s + 1.33e3·29-s + 7.99e3·31-s + 7.00e3·32-s + 1.84e4·34-s + 1.97e3·37-s − 1.07e4·38-s − 1.07e4·41-s − 1.50e4·43-s + 3.56e4·44-s + 7.04e3·46-s + 895.·47-s − 1.20e4·49-s + 3.14e4·52-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 2.29·4-s − 0.531·7-s + 2.34·8-s + 1.21·11-s + 0.703·13-s − 0.964·14-s + 1.95·16-s + 1.51·17-s − 0.665·19-s + 2.19·22-s + 0.270·23-s + 1.27·26-s − 1.21·28-s + 0.295·29-s + 1.49·31-s + 1.20·32-s + 2.74·34-s + 0.236·37-s − 1.20·38-s − 1.00·41-s − 1.23·43-s + 2.77·44-s + 0.491·46-s + 0.0591·47-s − 0.717·49-s + 1.61·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.739727904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.739727904\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 10.2T + 32T^{2} \) |
| 7 | \( 1 + 68.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 486.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 428.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 686.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.50e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 895.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.71e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.14e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5T + 8.58e9T^{2} \) |
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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
−11.84284378110971098592199863948, −10.80077650957365366675933423155, −9.647719812442375702280286682633, −8.139352601250493605267297460261, −6.67435433125964995025385835629, −6.21195657197929147311612400322, −4.97858512140980624268155246724, −3.83784300741976244371082874616, −3.05564640871666463783661388852, −1.41271695961258412948308310885,
1.41271695961258412948308310885, 3.05564640871666463783661388852, 3.83784300741976244371082874616, 4.97858512140980624268155246724, 6.21195657197929147311612400322, 6.67435433125964995025385835629, 8.139352601250493605267297460261, 9.647719812442375702280286682633, 10.80077650957365366675933423155, 11.84284378110971098592199863948
