| L(s) = 1 | − 3-s + 9-s − 5.65·11-s − 5.65·13-s − 5.65·17-s + 4·19-s − 5.65·23-s − 5·25-s − 27-s − 6·29-s + 8·31-s + 5.65·33-s + 2·37-s + 5.65·39-s + 5.65·41-s + 8·47-s + 5.65·51-s − 2·53-s − 4·57-s + 4·59-s − 5.65·61-s + 11.3·67-s + 5.65·69-s + 5.65·71-s + 11.3·73-s + 5·75-s + 11.3·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.333·9-s − 1.70·11-s − 1.56·13-s − 1.37·17-s + 0.917·19-s − 1.17·23-s − 25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.984·33-s + 0.328·37-s + 0.905·39-s + 0.883·41-s + 1.16·47-s + 0.792·51-s − 0.274·53-s − 0.529·57-s + 0.520·59-s − 0.724·61-s + 1.38·67-s + 0.681·69-s + 0.671·71-s + 1.32·73-s + 0.577·75-s + 1.27·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6461332839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6461332839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 5.65T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
| 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
−7.918612022604341217808372033098, −7.73674952388092284124541673619, −6.90255010493920714418944634150, −6.00488743153619097652152661056, −5.32485468421909927535789492452, −4.74988340245842819534936233202, −3.93352631022221744166972365296, −2.59824645629894592600782620378, −2.16112159249811012191308944914, −0.42626342889688379898771070604,
0.42626342889688379898771070604, 2.16112159249811012191308944914, 2.59824645629894592600782620378, 3.93352631022221744166972365296, 4.74988340245842819534936233202, 5.32485468421909927535789492452, 6.00488743153619097652152661056, 6.90255010493920714418944634150, 7.73674952388092284124541673619, 7.918612022604341217808372033098
