| L(s) = 1 | + 5.98·2-s − 30.4·3-s + 3.87·4-s + 25·5-s − 182.·6-s − 221.·7-s − 168.·8-s + 685.·9-s + 149.·10-s + 121·11-s − 118.·12-s − 152.·13-s − 1.32e3·14-s − 761.·15-s − 1.13e3·16-s + 247.·17-s + 4.10e3·18-s − 361·19-s + 96.9·20-s + 6.73e3·21-s + 724.·22-s − 3.53e3·23-s + 5.13e3·24-s + 625·25-s − 912.·26-s − 1.34e4·27-s − 857.·28-s + ⋯ |
| L(s) = 1 | + 1.05·2-s − 1.95·3-s + 0.121·4-s + 0.447·5-s − 2.07·6-s − 1.70·7-s − 0.930·8-s + 2.82·9-s + 0.473·10-s + 0.301·11-s − 0.236·12-s − 0.249·13-s − 1.80·14-s − 0.874·15-s − 1.10·16-s + 0.207·17-s + 2.98·18-s − 0.229·19-s + 0.0542·20-s + 3.33·21-s + 0.319·22-s − 1.39·23-s + 1.81·24-s + 0.200·25-s − 0.264·26-s − 3.56·27-s − 0.206·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 5.98T + 32T^{2} \) |
| 3 | \( 1 + 30.4T + 243T^{2} \) |
| 7 | \( 1 + 221.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 152.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 247.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.06e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 753.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.97e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.24e5T + 8.58e9T^{2} \) |
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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
−9.206887215049352143037565793862, −7.41001802270054451613863842796, −6.43833731165166445283292020143, −6.05257926017788600520922770754, −5.59662148098882375804491169597, −4.43409130743552150315259730252, −3.90068705007148144549916454424, −2.54385206824437318903557210457, −0.842184289394915476364222416611, 0,
0.842184289394915476364222416611, 2.54385206824437318903557210457, 3.90068705007148144549916454424, 4.43409130743552150315259730252, 5.59662148098882375804491169597, 6.05257926017788600520922770754, 6.43833731165166445283292020143, 7.41001802270054451613863842796, 9.206887215049352143037565793862
