| L(s) = 1 | − 10.8·2-s − 20.9·3-s + 86.0·4-s − 25·5-s + 227.·6-s + 150.·7-s − 587.·8-s + 195.·9-s + 271.·10-s − 1.80e3·12-s − 871.·13-s − 1.63e3·14-s + 523.·15-s + 3.62e3·16-s − 650.·17-s − 2.12e3·18-s + 403.·19-s − 2.15e3·20-s − 3.14e3·21-s − 3.56e3·23-s + 1.22e4·24-s + 625·25-s + 9.47e3·26-s + 999.·27-s + 1.29e4·28-s + 685.·29-s − 5.68e3·30-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 1.34·3-s + 2.68·4-s − 0.447·5-s + 2.57·6-s + 1.15·7-s − 3.24·8-s + 0.803·9-s + 0.859·10-s − 3.61·12-s − 1.43·13-s − 2.22·14-s + 0.600·15-s + 3.54·16-s − 0.546·17-s − 1.54·18-s + 0.256·19-s − 1.20·20-s − 1.55·21-s − 1.40·23-s + 4.35·24-s + 0.200·25-s + 2.74·26-s + 0.263·27-s + 3.11·28-s + 0.151·29-s − 1.15·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1913060419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1913060419\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 10.8T + 32T^{2} \) |
| 3 | \( 1 + 20.9T + 243T^{2} \) |
| 7 | \( 1 - 150.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 871.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 650.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 403.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 685.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 94.4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.53e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 692.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.92e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.24e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.37e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e5T + 8.58e9T^{2} \) |
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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
−9.928598733870768416759623644946, −9.092272090709199734345192213889, −7.947589632134820232734172940107, −7.55535556173414596961187546661, −6.56373207855064062965942584337, −5.62282719106485167623458349711, −4.51483603724326880915554683907, −2.50432255630746588632670237540, −1.43962910773478378700475500710, −0.31659253051161526580919351371,
0.31659253051161526580919351371, 1.43962910773478378700475500710, 2.50432255630746588632670237540, 4.51483603724326880915554683907, 5.62282719106485167623458349711, 6.56373207855064062965942584337, 7.55535556173414596961187546661, 7.947589632134820232734172940107, 9.092272090709199734345192213889, 9.928598733870768416759623644946
