| L(s) = 1 | − 8.50·2-s + 40.3·4-s + 86.6·5-s − 64.0·7-s − 71.0·8-s − 737.·10-s + 285.·11-s − 307.·13-s + 544.·14-s − 686.·16-s − 2.22e3·17-s − 1.80e3·19-s + 3.49e3·20-s − 2.42e3·22-s + 529·23-s + 4.39e3·25-s + 2.61e3·26-s − 2.58e3·28-s + 2.35e3·29-s + 8.31e3·31-s + 8.11e3·32-s + 1.89e4·34-s − 5.54e3·35-s − 9.07e3·37-s + 1.53e4·38-s − 6.15e3·40-s − 1.54e3·41-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 1.26·4-s + 1.55·5-s − 0.493·7-s − 0.392·8-s − 2.33·10-s + 0.710·11-s − 0.504·13-s + 0.742·14-s − 0.670·16-s − 1.86·17-s − 1.14·19-s + 1.95·20-s − 1.06·22-s + 0.208·23-s + 1.40·25-s + 0.758·26-s − 0.622·28-s + 0.520·29-s + 1.55·31-s + 1.40·32-s + 2.80·34-s − 0.765·35-s − 1.08·37-s + 1.72·38-s − 0.608·40-s − 0.143·41-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 8.50T + 32T^{2} \) |
| 5 | \( 1 - 86.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 64.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 285.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 307.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.80e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.54e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.53e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.47e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.40e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.61e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.00e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.17e4T + 8.58e9T^{2} \) |
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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
−10.49523007256810870932015823689, −9.954751953721200765611179335581, −9.082836120843059806028684283992, −8.492156869425923832689506267852, −6.74899554809200470118325426402, −6.42944620424172418297063069113, −4.68454306429594043327677136536, −2.48685077181641424121250467294, −1.56838595786252718871712940098, 0,
1.56838595786252718871712940098, 2.48685077181641424121250467294, 4.68454306429594043327677136536, 6.42944620424172418297063069113, 6.74899554809200470118325426402, 8.492156869425923832689506267852, 9.082836120843059806028684283992, 9.954751953721200765611179335581, 10.49523007256810870932015823689
