L(s) = 1 | − 2-s − 1.86·3-s + 4-s − 0.189·5-s + 1.86·6-s − 4.05·7-s − 8-s + 0.471·9-s + 0.189·10-s − 11-s − 1.86·12-s + 2.20·13-s + 4.05·14-s + 0.353·15-s + 16-s − 2.49·17-s − 0.471·18-s + 3.29·19-s − 0.189·20-s + 7.54·21-s + 22-s + 2.66·23-s + 1.86·24-s − 4.96·25-s − 2.20·26-s + 4.71·27-s − 4.05·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.07·3-s + 0.5·4-s − 0.0847·5-s + 0.760·6-s − 1.53·7-s − 0.353·8-s + 0.157·9-s + 0.0599·10-s − 0.301·11-s − 0.537·12-s + 0.611·13-s + 1.08·14-s + 0.0912·15-s + 0.250·16-s − 0.604·17-s − 0.111·18-s + 0.756·19-s − 0.0423·20-s + 1.64·21-s + 0.213·22-s + 0.556·23-s + 0.380·24-s − 0.992·25-s − 0.432·26-s + 0.906·27-s − 0.765·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
good | 3 | \( 1 + 1.86T + 3T^{2} \) |
| 5 | \( 1 + 0.189T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 - 0.251T + 31T^{2} \) |
| 37 | \( 1 - 7.30T + 37T^{2} \) |
| 41 | \( 1 - 3.69T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.06T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 - 4.97T + 67T^{2} \) |
| 71 | \( 1 - 5.09T + 71T^{2} \) |
| 73 | \( 1 - 7.97T + 73T^{2} \) |
| 79 | \( 1 + 1.72T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 3.84T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−7.936445393654496996052523703415, −7.18745093282025225469005263508, −6.51630773846702734097498016214, −5.91113016094717690379330690472, −5.42451760612721858173791063454, −4.16145694394435043047600564870, −3.29179963708481230734053001480, −2.39698625252297095377258898750, −0.912859137507304723934639991040, 0,
0.912859137507304723934639991040, 2.39698625252297095377258898750, 3.29179963708481230734053001480, 4.16145694394435043047600564870, 5.42451760612721858173791063454, 5.91113016094717690379330690472, 6.51630773846702734097498016214, 7.18745093282025225469005263508, 7.936445393654496996052523703415