| L(s) = 1 | + 2.42·2-s + 1.34·3-s − 2.12·4-s − 5·5-s + 3.27·6-s − 30.7·7-s − 24.5·8-s − 25.1·9-s − 12.1·10-s − 41.4·11-s − 2.86·12-s − 82.0·13-s − 74.4·14-s − 6.74·15-s − 42.4·16-s − 61.0·18-s + 82.3·19-s + 10.6·20-s − 41.4·21-s − 100.·22-s − 27.7·23-s − 33.1·24-s + 25·25-s − 198.·26-s − 70.4·27-s + 65.2·28-s + 139.·29-s + ⋯ |
| L(s) = 1 | + 0.856·2-s + 0.259·3-s − 0.265·4-s − 0.447·5-s + 0.222·6-s − 1.65·7-s − 1.08·8-s − 0.932·9-s − 0.383·10-s − 1.13·11-s − 0.0689·12-s − 1.74·13-s − 1.42·14-s − 0.116·15-s − 0.663·16-s − 0.799·18-s + 0.994·19-s + 0.118·20-s − 0.430·21-s − 0.974·22-s − 0.251·23-s − 0.281·24-s + 0.200·25-s − 1.49·26-s − 0.501·27-s + 0.440·28-s + 0.895·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04394783056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04394783056\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.42T + 8T^{2} \) |
| 3 | \( 1 - 1.34T + 27T^{2} \) |
| 7 | \( 1 + 30.7T + 343T^{2} \) |
| 11 | \( 1 + 41.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82.0T + 2.19e3T^{2} \) |
| 19 | \( 1 - 82.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 198.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 481.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 130.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 168.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 201.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 362.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 771.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 522.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 123.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 191.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.31e3T + 9.12e5T^{2} \) |
| 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
−9.370813552884377433293360815322, −8.340666850783060313333318498233, −7.46520997069675294397756702574, −6.62439673140533588924886057831, −5.55474418482597299103682367099, −5.11775823597262044005187723643, −3.93598076723090669531476748421, −2.96876699084541972083572979410, −2.72504942671428090292500234860, −0.081398477882163589680451254129,
0.081398477882163589680451254129, 2.72504942671428090292500234860, 2.96876699084541972083572979410, 3.93598076723090669531476748421, 5.11775823597262044005187723643, 5.55474418482597299103682367099, 6.62439673140533588924886057831, 7.46520997069675294397756702574, 8.340666850783060313333318498233, 9.370813552884377433293360815322
