| L(s) = 1 | − 8.18·2-s + 35.0·4-s + 59.8·5-s + 145.·7-s − 24.8·8-s − 490.·10-s − 121·11-s + 615.·13-s − 1.18e3·14-s − 917.·16-s − 1.84e3·17-s + 366.·19-s + 2.09e3·20-s + 990.·22-s + 4.51e3·23-s + 459.·25-s − 5.04e3·26-s + 5.08e3·28-s + 1.71e3·29-s − 2.65e3·31-s + 8.30e3·32-s + 1.50e4·34-s + 8.68e3·35-s + 9.66e3·37-s − 3.00e3·38-s − 1.48e3·40-s + 1.11e4·41-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.09·4-s + 1.07·5-s + 1.11·7-s − 0.137·8-s − 1.55·10-s − 0.301·11-s + 1.01·13-s − 1.61·14-s − 0.896·16-s − 1.54·17-s + 0.232·19-s + 1.17·20-s + 0.436·22-s + 1.78·23-s + 0.147·25-s − 1.46·26-s + 1.22·28-s + 0.379·29-s − 0.495·31-s + 1.43·32-s + 2.23·34-s + 1.19·35-s + 1.16·37-s − 0.337·38-s − 0.147·40-s + 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.223636672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223636672\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 8.18T + 32T^{2} \) |
| 5 | \( 1 - 59.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 145.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 615.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.84e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 366.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.39e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.83e4T + 8.58e9T^{2} \) |
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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
−13.09044133691739861670882075988, −11.11788636040633740158406670087, −10.85816404567449948877611221452, −9.425850760959062506184125156646, −8.771647902938532080187999640549, −7.66615299119126929011214717355, −6.33445668466258336422048453654, −4.78997703276040416983981787911, −2.22068982757566119899778413809, −1.07359678769425008406181978019,
1.07359678769425008406181978019, 2.22068982757566119899778413809, 4.78997703276040416983981787911, 6.33445668466258336422048453654, 7.66615299119126929011214717355, 8.771647902938532080187999640549, 9.425850760959062506184125156646, 10.85816404567449948877611221452, 11.11788636040633740158406670087, 13.09044133691739861670882075988
