| L(s) = 1 | − 9.60·2-s + 21.6·3-s + 60.2·4-s − 25·5-s − 207.·6-s − 51.1·7-s − 271.·8-s + 223.·9-s + 240.·10-s + 1.30e3·12-s − 495.·13-s + 491.·14-s − 540.·15-s + 681.·16-s − 155.·17-s − 2.14e3·18-s + 181.·19-s − 1.50e3·20-s − 1.10e3·21-s + 3.72e3·23-s − 5.86e3·24-s + 625·25-s + 4.75e3·26-s − 419.·27-s − 3.08e3·28-s − 5.21e3·29-s + 5.18e3·30-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.38·3-s + 1.88·4-s − 0.447·5-s − 2.35·6-s − 0.394·7-s − 1.50·8-s + 0.920·9-s + 0.759·10-s + 2.61·12-s − 0.812·13-s + 0.670·14-s − 0.619·15-s + 0.665·16-s − 0.130·17-s − 1.56·18-s + 0.115·19-s − 0.842·20-s − 0.546·21-s + 1.46·23-s − 2.08·24-s + 0.200·25-s + 1.37·26-s − 0.110·27-s − 0.743·28-s − 1.15·29-s + 1.05·30-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.60T + 32T^{2} \) |
| 3 | \( 1 - 21.6T + 243T^{2} \) |
| 7 | \( 1 + 51.1T + 1.68e4T^{2} \) |
| 13 | \( 1 + 495.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 155.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 181.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.72e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.66e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.69e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.86e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 980.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.10e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.47e5T + 8.58e9T^{2} \) |
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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
−9.420140679529332289505498253504, −8.642915239116640184387742291102, −7.83022500536864379586930216982, −7.39712953019000425263412133613, −6.36267559289189473904757749227, −4.55812715747432299586347177773, −3.11841885563483704923936913501, −2.48564598623346511984361481987, −1.23455740042346775826500278631, 0,
1.23455740042346775826500278631, 2.48564598623346511984361481987, 3.11841885563483704923936913501, 4.55812715747432299586347177773, 6.36267559289189473904757749227, 7.39712953019000425263412133613, 7.83022500536864379586930216982, 8.642915239116640184387742291102, 9.420140679529332289505498253504
