| L(s) = 1 | − i·5-s + 3.18i·7-s − 2.46·11-s + 1.87·13-s − 4.53i·17-s + 6.27i·19-s + 0.551·23-s − 25-s − 4.88i·29-s + 5.26i·31-s + 3.18·35-s − 5.30·37-s + 8.02i·41-s − 7.54i·43-s + 3.90·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.20i·7-s − 0.741·11-s + 0.519·13-s − 1.09i·17-s + 1.44i·19-s + 0.115·23-s − 0.200·25-s − 0.906i·29-s + 0.945i·31-s + 0.537·35-s − 0.872·37-s + 1.25i·41-s − 1.15i·43-s + 0.569·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8702123633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8702123633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.18iT - 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 + 4.53iT - 17T^{2} \) |
| 19 | \( 1 - 6.27iT - 19T^{2} \) |
| 23 | \( 1 - 0.551T + 23T^{2} \) |
| 29 | \( 1 + 4.88iT - 29T^{2} \) |
| 31 | \( 1 - 5.26iT - 31T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 - 8.02iT - 41T^{2} \) |
| 43 | \( 1 + 7.54iT - 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 - 8.82iT - 53T^{2} \) |
| 59 | \( 1 + 5.06T + 59T^{2} \) |
| 61 | \( 1 - 0.587T + 61T^{2} \) |
| 67 | \( 1 - 6.54iT - 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 + 6.65T + 73T^{2} \) |
| 79 | \( 1 + 3.21iT - 79T^{2} \) |
| 83 | \( 1 + 9.38T + 83T^{2} \) |
| 89 | \( 1 - 4.99iT - 89T^{2} \) |
| 97 | \( 1 - 8.03T + 97T^{2} \) |
| 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
−8.679942865311905119685769804240, −8.026515347470640171517439146548, −7.32205770168370246888545157435, −6.28258051246571094908882565575, −5.61644639254628830032802684213, −5.13187598808562115555487678156, −4.17500763901287367825540810880, −3.13347110443748639765758005737, −2.37427211598805336934755581693, −1.31662756588333694023226409697,
0.24509826919234156132380620605, 1.49843687503087009036895004677, 2.64780898736161015350324082957, 3.54364093060680859973173298758, 4.22804526765214506199075373006, 5.09194904974691553895522895023, 5.97356547705069885130679192251, 6.81384655667784548718294716390, 7.28040471428961640321525094359, 8.038207629256550118608743098910
