| L(s) = 1 | − 2-s − 1.45·3-s + 4-s + 0.421·5-s + 1.45·6-s − 7-s − 8-s − 0.883·9-s − 0.421·10-s + 5.56·11-s − 1.45·12-s − 6.08·13-s + 14-s − 0.612·15-s + 16-s + 1.25·17-s + 0.883·18-s + 2.04·19-s + 0.421·20-s + 1.45·21-s − 5.56·22-s + 5.01·23-s + 1.45·24-s − 4.82·25-s + 6.08·26-s + 5.64·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.839·3-s + 0.5·4-s + 0.188·5-s + 0.593·6-s − 0.377·7-s − 0.353·8-s − 0.294·9-s − 0.133·10-s + 1.67·11-s − 0.419·12-s − 1.68·13-s + 0.267·14-s − 0.158·15-s + 0.250·16-s + 0.303·17-s + 0.208·18-s + 0.468·19-s + 0.0942·20-s + 0.317·21-s − 1.18·22-s + 1.04·23-s + 0.296·24-s − 0.964·25-s + 1.19·26-s + 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 - 0.421T + 5T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 6.08T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 - 5.01T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 41 | \( 1 + 8.49T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 - 7.91T + 47T^{2} \) |
| 53 | \( 1 - 2.13T + 53T^{2} \) |
| 59 | \( 1 - 5.95T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 3.25T + 67T^{2} \) |
| 71 | \( 1 - 9.78T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 + 9.27T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 0.964T + 89T^{2} \) |
| 97 | \( 1 - 5.02T + 97T^{2} \) |
| 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
−7.48205384713921648213733521870, −7.07233673379435364741657895563, −6.40985113130014328243746659788, −5.64420688101175894999761252612, −5.09723882603439222420109106588, −4.02589446426563472183826886685, −3.12390800265932914506074863294, −2.13352039723551708846695316807, −1.06853905866146379247836015864, 0,
1.06853905866146379247836015864, 2.13352039723551708846695316807, 3.12390800265932914506074863294, 4.02589446426563472183826886685, 5.09723882603439222420109106588, 5.64420688101175894999761252612, 6.40985113130014328243746659788, 7.07233673379435364741657895563, 7.48205384713921648213733521870
