| L(s) = 1 | + 1.49·2-s + 1.05·3-s + 0.247·4-s − 0.565·5-s + 1.57·6-s − 4.58·7-s − 2.62·8-s − 1.89·9-s − 0.848·10-s + 0.260·12-s + 2.43·13-s − 6.87·14-s − 0.595·15-s − 4.43·16-s − 1.31·17-s − 2.83·18-s − 6.64·19-s − 0.139·20-s − 4.83·21-s + 7.72·23-s − 2.76·24-s − 4.67·25-s + 3.65·26-s − 5.15·27-s − 1.13·28-s + 1.45·29-s − 0.893·30-s + ⋯ |
| L(s) = 1 | + 1.05·2-s + 0.608·3-s + 0.123·4-s − 0.253·5-s + 0.644·6-s − 1.73·7-s − 0.928·8-s − 0.630·9-s − 0.268·10-s + 0.0751·12-s + 0.676·13-s − 1.83·14-s − 0.153·15-s − 1.10·16-s − 0.318·17-s − 0.668·18-s − 1.52·19-s − 0.0312·20-s − 1.05·21-s + 1.61·23-s − 0.564·24-s − 0.935·25-s + 0.716·26-s − 0.991·27-s − 0.214·28-s + 0.270·29-s − 0.163·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804917240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804917240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.49T + 2T^{2} \) |
| 3 | \( 1 - 1.05T + 3T^{2} \) |
| 5 | \( 1 + 0.565T + 5T^{2} \) |
| 7 | \( 1 + 4.58T + 7T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 + 6.64T + 19T^{2} \) |
| 23 | \( 1 - 7.72T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 5.69T + 37T^{2} \) |
| 41 | \( 1 - 5.83T + 41T^{2} \) |
| 43 | \( 1 + 4.33T + 43T^{2} \) |
| 47 | \( 1 + 0.120T + 47T^{2} \) |
| 53 | \( 1 - 2.59T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 + 3.39T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
| 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
−8.028795382977772541504490947268, −6.77051558330148869182775392910, −6.48709157283255958643415841817, −5.85284209969468048129191598573, −5.01793803913534277663658656260, −4.06409472262753654378710868218, −3.62097437106085422772559217666, −2.92297454528705585925136294030, −2.38849406729045324379685978966, −0.52142529505512525403349970461,
0.52142529505512525403349970461, 2.38849406729045324379685978966, 2.92297454528705585925136294030, 3.62097437106085422772559217666, 4.06409472262753654378710868218, 5.01793803913534277663658656260, 5.85284209969468048129191598573, 6.48709157283255958643415841817, 6.77051558330148869182775392910, 8.028795382977772541504490947268
