L(s) = 1 | − 2.80·2-s + 3-s + 5.88·4-s − 2.90·5-s − 2.80·6-s − 1.28·7-s − 10.9·8-s + 9-s + 8.16·10-s + 2.87·11-s + 5.88·12-s + 0.133·13-s + 3.61·14-s − 2.90·15-s + 18.8·16-s − 17-s − 2.80·18-s − 3.66·19-s − 17.1·20-s − 1.28·21-s − 8.06·22-s − 6.74·23-s − 10.9·24-s + 3.45·25-s − 0.374·26-s + 27-s − 7.58·28-s + ⋯ |
L(s) = 1 | − 1.98·2-s + 0.577·3-s + 2.94·4-s − 1.30·5-s − 1.14·6-s − 0.486·7-s − 3.85·8-s + 0.333·9-s + 2.58·10-s + 0.865·11-s + 1.69·12-s + 0.0369·13-s + 0.966·14-s − 0.750·15-s + 4.71·16-s − 0.242·17-s − 0.661·18-s − 0.839·19-s − 3.82·20-s − 0.281·21-s − 1.71·22-s − 1.40·23-s − 2.22·24-s + 0.691·25-s − 0.0733·26-s + 0.192·27-s − 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 + 1.28T + 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 - 0.133T + 13T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 + 3.28T + 29T^{2} \) |
| 31 | \( 1 - 8.99T + 31T^{2} \) |
| 37 | \( 1 - 2.72T + 37T^{2} \) |
| 41 | \( 1 - 6.71T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 - 3.31T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 + 5.26T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 9.57T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 8.78T + 97T^{2} \) |
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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
−7.79990645351245544122566793245, −7.12410228790419701343655695605, −6.47656329963373076970938067350, −5.96124928636607966204755751024, −4.27515999038111862897060547484, −3.69256463691033108603300688063, −2.80278770230908203118817917176, −2.04249970815033471440505177340, −0.959414170124880857611425430628, 0,
0.959414170124880857611425430628, 2.04249970815033471440505177340, 2.80278770230908203118817917176, 3.69256463691033108603300688063, 4.27515999038111862897060547484, 5.96124928636607966204755751024, 6.47656329963373076970938067350, 7.12410228790419701343655695605, 7.79990645351245544122566793245