| L(s) = 1 | − 4.15·2-s − 29.2·3-s − 14.7·4-s + 121.·6-s − 42.5·7-s + 194.·8-s + 611.·9-s + 434.·11-s + 431.·12-s + 169·13-s + 176.·14-s − 334.·16-s + 424.·17-s − 2.53e3·18-s − 2.20e3·19-s + 1.24e3·21-s − 1.80e3·22-s + 1.17e3·23-s − 5.67e3·24-s − 701.·26-s − 1.07e4·27-s + 627.·28-s − 3.07e3·29-s + 6.24e3·31-s − 4.82e3·32-s − 1.26e4·33-s − 1.76e3·34-s + ⋯ |
| L(s) = 1 | − 0.734·2-s − 1.87·3-s − 0.460·4-s + 1.37·6-s − 0.327·7-s + 1.07·8-s + 2.51·9-s + 1.08·11-s + 0.864·12-s + 0.277·13-s + 0.240·14-s − 0.326·16-s + 0.356·17-s − 1.84·18-s − 1.40·19-s + 0.614·21-s − 0.794·22-s + 0.463·23-s − 2.01·24-s − 0.203·26-s − 2.84·27-s + 0.151·28-s − 0.678·29-s + 1.16·31-s − 0.832·32-s − 2.02·33-s − 0.261·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4782944625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4782944625\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 4.15T + 32T^{2} \) |
| 3 | \( 1 + 29.2T + 243T^{2} \) |
| 7 | \( 1 + 42.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 434.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 424.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.20e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.08e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.61e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.63e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.81e5T + 8.58e9T^{2} \) |
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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
−10.82090266398215546494364654965, −9.912427748856147049915997741165, −9.191066306066787366268832992116, −7.88620589004347573383806346323, −6.69034554632166940905699142458, −6.05905458760344052876356385412, −4.81043956800408024057518885081, −4.03169695043063648149652713051, −1.45201455924205559157608851914, −0.52474212714070350780990581765,
0.52474212714070350780990581765, 1.45201455924205559157608851914, 4.03169695043063648149652713051, 4.81043956800408024057518885081, 6.05905458760344052876356385412, 6.69034554632166940905699142458, 7.88620589004347573383806346323, 9.191066306066787366268832992116, 9.912427748856147049915997741165, 10.82090266398215546494364654965
