L(s) = 1 | − 1.69·2-s − 3-s + 0.873·4-s + 5-s + 1.69·6-s − 0.214·7-s + 1.90·8-s + 9-s − 1.69·10-s − 0.624·11-s − 0.873·12-s + 3.83·13-s + 0.364·14-s − 15-s − 4.98·16-s − 3.68·17-s − 1.69·18-s − 0.920·19-s + 0.873·20-s + 0.214·21-s + 1.05·22-s + 1.09·23-s − 1.90·24-s + 25-s − 6.49·26-s − 27-s − 0.187·28-s + ⋯ |
L(s) = 1 | − 1.19·2-s − 0.577·3-s + 0.436·4-s + 0.447·5-s + 0.692·6-s − 0.0812·7-s + 0.674·8-s + 0.333·9-s − 0.536·10-s − 0.188·11-s − 0.252·12-s + 1.06·13-s + 0.0973·14-s − 0.258·15-s − 1.24·16-s − 0.892·17-s − 0.399·18-s − 0.211·19-s + 0.195·20-s + 0.0469·21-s + 0.225·22-s + 0.227·23-s − 0.389·24-s + 0.200·25-s − 1.27·26-s − 0.192·27-s − 0.0354·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 7 | \( 1 + 0.214T + 7T^{2} \) |
| 11 | \( 1 + 0.624T + 11T^{2} \) |
| 13 | \( 1 - 3.83T + 13T^{2} \) |
| 17 | \( 1 + 3.68T + 17T^{2} \) |
| 19 | \( 1 + 0.920T + 19T^{2} \) |
| 23 | \( 1 - 1.09T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 + 8.43T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 0.669T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 + 7.65T + 67T^{2} \) |
| 71 | \( 1 + 0.452T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.110317400930113790762588190695, −6.82519922407085080377456694919, −6.65097724535229722952119785893, −5.72558336009851843245285376308, −4.84021201988954801794019769195, −4.21683191351825824748643290841, −3.03406027111032792441199563807, −1.89485922931472212634797181406, −1.13342395372241437118501415087, 0,
1.13342395372241437118501415087, 1.89485922931472212634797181406, 3.03406027111032792441199563807, 4.21683191351825824748643290841, 4.84021201988954801794019769195, 5.72558336009851843245285376308, 6.65097724535229722952119785893, 6.82519922407085080377456694919, 8.110317400930113790762588190695