Error: no document with id 250969025 found in table mf_hecke_traces.
Error: table True does not exist
| L(s) = 1 | − 0.328·2-s − 1.89·4-s + 1.86·7-s + 1.27·8-s + 0.564·11-s + 4.95·13-s − 0.613·14-s + 3.36·16-s + 7.06·17-s − 3.32·19-s − 0.185·22-s + 2.36·23-s − 1.63·26-s − 3.52·28-s − 29-s + 4.87·31-s − 3.66·32-s − 2.32·34-s + 8.50·37-s + 1.09·38-s + 7.86·41-s − 4.39·43-s − 1.06·44-s − 0.777·46-s + 7.06·47-s − 3.51·49-s − 9.38·52-s + ⋯ |
| L(s) = 1 | − 0.232·2-s − 0.945·4-s + 0.705·7-s + 0.452·8-s + 0.170·11-s + 1.37·13-s − 0.164·14-s + 0.840·16-s + 1.71·17-s − 0.763·19-s − 0.0395·22-s + 0.493·23-s − 0.319·26-s − 0.667·28-s − 0.185·29-s + 0.875·31-s − 0.648·32-s − 0.398·34-s + 1.39·37-s + 0.177·38-s + 1.22·41-s − 0.670·43-s − 0.160·44-s − 0.114·46-s + 1.03·47-s − 0.502·49-s − 1.30·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.920615096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.920615096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.328T + 2T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 - 0.564T + 11T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 - 7.06T + 17T^{2} \) |
| 19 | \( 1 + 3.32T + 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 - 8.50T + 37T^{2} \) |
| 41 | \( 1 - 7.86T + 41T^{2} \) |
| 43 | \( 1 + 4.39T + 43T^{2} \) |
| 47 | \( 1 - 7.06T + 47T^{2} \) |
| 53 | \( 1 - 4.18T + 53T^{2} \) |
| 59 | \( 1 + 9.01T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 - 6.61T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 + 0.622T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.969147825613377091341383631995, −7.71789174672567345921912851457, −6.50458780712665745826960047607, −5.81439394832548118728484934582, −5.16551537337584141371189825724, −4.31612470847435000036925208612, −3.78994364230606449456248842583, −2.83927542959698702178281276160, −1.45663227524667777664408495206, −0.860499368800430087433976880762,
0.860499368800430087433976880762, 1.45663227524667777664408495206, 2.83927542959698702178281276160, 3.78994364230606449456248842583, 4.31612470847435000036925208612, 5.16551537337584141371189825724, 5.81439394832548118728484934582, 6.50458780712665745826960047607, 7.71789174672567345921912851457, 7.969147825613377091341383631995
