| L(s) = 1 | + 3-s + 1.09·5-s − 1.62·7-s + 9-s − 4.16·11-s + 3.95·13-s + 1.09·15-s − 2.36·17-s + 4.37·19-s − 1.62·21-s − 23-s − 3.79·25-s + 27-s − 29-s − 8.09·31-s − 4.16·33-s − 1.78·35-s + 3.52·37-s + 3.95·39-s − 3.81·41-s + 1.25·43-s + 1.09·45-s + 6.41·47-s − 4.36·49-s − 2.36·51-s − 3.55·53-s − 4.57·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.491·5-s − 0.613·7-s + 0.333·9-s − 1.25·11-s + 1.09·13-s + 0.283·15-s − 0.573·17-s + 1.00·19-s − 0.354·21-s − 0.208·23-s − 0.758·25-s + 0.192·27-s − 0.185·29-s − 1.45·31-s − 0.724·33-s − 0.301·35-s + 0.580·37-s + 0.633·39-s − 0.595·41-s + 0.190·43-s + 0.163·45-s + 0.935·47-s − 0.623·49-s − 0.331·51-s − 0.488·53-s − 0.617·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 - 1.25T + 43T^{2} \) |
| 47 | \( 1 - 6.41T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 8.30T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 + 4.52T + 89T^{2} \) |
| 97 | \( 1 + 2.79T + 97T^{2} \) |
| 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
−7.57249071415762508687721535552, −6.87925980153016012551905104307, −5.95421362455706073912675742998, −5.60225512718419797169358112222, −4.63528468081093853411339613268, −3.71643343231114502212107616779, −3.10246371787541326504620773252, −2.30290215905754317137563645714, −1.43074786918120363678817831060, 0,
1.43074786918120363678817831060, 2.30290215905754317137563645714, 3.10246371787541326504620773252, 3.71643343231114502212107616779, 4.63528468081093853411339613268, 5.60225512718419797169358112222, 5.95421362455706073912675742998, 6.87925980153016012551905104307, 7.57249071415762508687721535552
