| L(s) = 1 | + 4.05·2-s − 0.841·3-s + 8.40·4-s + 17.3·5-s − 3.40·6-s + 1.64·8-s − 26.2·9-s + 70.0·10-s + 64.6·11-s − 7.07·12-s − 30.9·13-s − 14.5·15-s − 60.5·16-s − 94.7·17-s − 106.·18-s − 72.9·19-s + 145.·20-s + 261.·22-s − 173.·23-s − 1.38·24-s + 174.·25-s − 125.·26-s + 44.8·27-s − 127.·29-s − 58.9·30-s − 316.·31-s − 258.·32-s + ⋯ |
| L(s) = 1 | + 1.43·2-s − 0.161·3-s + 1.05·4-s + 1.54·5-s − 0.231·6-s + 0.0725·8-s − 0.973·9-s + 2.21·10-s + 1.77·11-s − 0.170·12-s − 0.659·13-s − 0.250·15-s − 0.946·16-s − 1.35·17-s − 1.39·18-s − 0.881·19-s + 1.62·20-s + 2.53·22-s − 1.56·23-s − 0.0117·24-s + 1.39·25-s − 0.945·26-s + 0.319·27-s − 0.818·29-s − 0.358·30-s − 1.83·31-s − 1.42·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 - 4.05T + 8T^{2} \) |
| 3 | \( 1 + 0.841T + 27T^{2} \) |
| 5 | \( 1 - 17.3T + 125T^{2} \) |
| 11 | \( 1 - 64.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 127.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 316.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 5.63T + 6.89e4T^{2} \) |
| 47 | \( 1 - 399.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 710.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 371.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 186.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 422.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 573.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 579.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 583.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.666409266505331234855683872444, −6.99991062010605316763465962296, −6.44354034664440874950564754941, −5.84582159313404238555844193189, −5.31672602126874730224355342021, −4.28225089863606070897602089768, −3.60346155041178611904708653376, −2.26454270227070968636203066623, −1.92973794829902050083479299028, 0,
1.92973794829902050083479299028, 2.26454270227070968636203066623, 3.60346155041178611904708653376, 4.28225089863606070897602089768, 5.31672602126874730224355342021, 5.84582159313404238555844193189, 6.44354034664440874950564754941, 6.99991062010605316763465962296, 8.666409266505331234855683872444
