| L(s) = 1 | + 8.45·2-s + 39.4·4-s − 58.5·5-s + 96.1·7-s + 63.0·8-s − 495.·10-s + 584.·11-s + 73.5·13-s + 812.·14-s − 729.·16-s + 1.72e3·17-s + 1.75e3·19-s − 2.31e3·20-s + 4.94e3·22-s + 529·23-s + 307.·25-s + 622.·26-s + 3.79e3·28-s + 6.00e3·29-s − 776.·31-s − 8.18e3·32-s + 1.46e4·34-s − 5.63e3·35-s + 1.47e4·37-s + 1.48e4·38-s − 3.69e3·40-s − 4.56e3·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.23·4-s − 1.04·5-s + 0.741·7-s + 0.348·8-s − 1.56·10-s + 1.45·11-s + 0.120·13-s + 1.10·14-s − 0.712·16-s + 1.44·17-s + 1.11·19-s − 1.29·20-s + 2.17·22-s + 0.208·23-s + 0.0984·25-s + 0.180·26-s + 0.914·28-s + 1.32·29-s − 0.145·31-s − 1.41·32-s + 2.16·34-s − 0.777·35-s + 1.77·37-s + 1.66·38-s − 0.364·40-s − 0.423·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.595727680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.595727680\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 8.45T + 32T^{2} \) |
| 5 | \( 1 + 58.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 96.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 584.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 73.5T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.72e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.75e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 776.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.63e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.59e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.56e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.29e4T + 8.58e9T^{2} \) |
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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
−11.87061703832221035799301110863, −11.18610379903968475786300096880, −9.596541210152858340046378417995, −8.259867312473065446087872710094, −7.25582449508229579146070351588, −6.08086238470917510911302535779, −4.91065225448781345320918518163, −4.00499051318781641216066894159, −3.13441799330995452030919693352, −1.17199697378019308956303658574,
1.17199697378019308956303658574, 3.13441799330995452030919693352, 4.00499051318781641216066894159, 4.91065225448781345320918518163, 6.08086238470917510911302535779, 7.25582449508229579146070351588, 8.259867312473065446087872710094, 9.596541210152858340046378417995, 11.18610379903968475786300096880, 11.87061703832221035799301110863
