| L(s) = 1 | − 1.87·3-s + 0.347·5-s + 4.75·7-s + 0.532·9-s − 4.75·11-s − 1.87·13-s − 0.652·15-s + 1.53·17-s − 8.94·21-s + 6.10·23-s − 4.87·25-s + 4.63·27-s − 1.98·29-s + 3.36·31-s + 8.94·33-s + 1.65·35-s − 3.38·37-s + 3.53·39-s − 4.49·41-s − 2.10·43-s + 0.184·45-s + 4.92·47-s + 15.6·49-s − 2.87·51-s − 12.5·53-s − 1.65·55-s − 5.66·59-s + ⋯ |
| L(s) = 1 | − 1.08·3-s + 0.155·5-s + 1.79·7-s + 0.177·9-s − 1.43·11-s − 0.521·13-s − 0.168·15-s + 0.371·17-s − 1.95·21-s + 1.27·23-s − 0.975·25-s + 0.892·27-s − 0.368·29-s + 0.605·31-s + 1.55·33-s + 0.279·35-s − 0.557·37-s + 0.565·39-s − 0.701·41-s − 0.321·43-s + 0.0275·45-s + 0.717·47-s + 2.23·49-s − 0.403·51-s − 1.72·53-s − 0.222·55-s − 0.736·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 0.347T + 5T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 + 1.98T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 - 9.99T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 5.63T + 89T^{2} \) |
| 97 | \( 1 - 7.66T + 97T^{2} \) |
| 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
−7.67860348324222142053989157665, −7.21251198257970835084985487540, −6.08602857203466359731765805539, −5.46898494989016865819234499716, −4.93228219239056951632296059735, −4.57651421114027559774947975046, −3.14497531647791671477894791960, −2.19416765775757011133889567482, −1.25224911698167647637461150590, 0,
1.25224911698167647637461150590, 2.19416765775757011133889567482, 3.14497531647791671477894791960, 4.57651421114027559774947975046, 4.93228219239056951632296059735, 5.46898494989016865819234499716, 6.08602857203466359731765805539, 7.21251198257970835084985487540, 7.67860348324222142053989157665
