| L(s) = 1 | − 1.48·2-s + 1.79·3-s + 0.208·4-s − 2.66·6-s + 1.48·7-s + 2.66·8-s + 0.208·9-s + 0.373·12-s + 5.63·13-s − 2.20·14-s − 4.37·16-s − 4.14·17-s − 0.310·18-s − 2.66·19-s + 2.66·21-s − 6.79·23-s + 4.76·24-s − 8.37·26-s − 5.00·27-s + 0.310·28-s − 10.0·29-s − 5·31-s + 1.17·32-s + 6.16·34-s + 0.0435·36-s − 8.58·37-s + 3.95·38-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 1.03·3-s + 0.104·4-s − 1.08·6-s + 0.561·7-s + 0.941·8-s + 0.0695·9-s + 0.107·12-s + 1.56·13-s − 0.590·14-s − 1.09·16-s − 1.00·17-s − 0.0731·18-s − 0.610·19-s + 0.580·21-s − 1.41·23-s + 0.973·24-s − 1.64·26-s − 0.962·27-s + 0.0586·28-s − 1.87·29-s − 0.898·31-s + 0.207·32-s + 1.05·34-s + 0.00726·36-s − 1.41·37-s + 0.641·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 + 6.79T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 43 | \( 1 - 2.97T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 6.79T + 53T^{2} \) |
| 59 | \( 1 - 6.16T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + 9.78T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 - 0.373T + 97T^{2} \) |
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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
−8.391613503537135884360342793474, −8.022105353916020317768017819143, −7.22966796612548019123865270705, −6.23668313702133705004103379402, −5.29513535885483983433532663765, −4.06891864224617506979556403757, −3.66632415427217356135060005273, −2.15226244202004724871599176430, −1.64361869270343592616466887494, 0,
1.64361869270343592616466887494, 2.15226244202004724871599176430, 3.66632415427217356135060005273, 4.06891864224617506979556403757, 5.29513535885483983433532663765, 6.23668313702133705004103379402, 7.22966796612548019123865270705, 8.022105353916020317768017819143, 8.391613503537135884360342793474
