| L(s) = 1 | − 2.69·2-s − 3-s + 5.28·4-s + 3.29·5-s + 2.69·6-s + 4.13·7-s − 8.87·8-s + 9-s − 8.89·10-s + 1.26·11-s − 5.28·12-s + 5.50·13-s − 11.1·14-s − 3.29·15-s + 13.3·16-s + 17-s − 2.69·18-s + 3.75·19-s + 17.4·20-s − 4.13·21-s − 3.41·22-s − 5.79·23-s + 8.87·24-s + 5.85·25-s − 14.8·26-s − 27-s + 21.8·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.577·3-s + 2.64·4-s + 1.47·5-s + 1.10·6-s + 1.56·7-s − 3.13·8-s + 0.333·9-s − 2.81·10-s + 0.381·11-s − 1.52·12-s + 1.52·13-s − 2.98·14-s − 0.850·15-s + 3.34·16-s + 0.242·17-s − 0.636·18-s + 0.860·19-s + 3.89·20-s − 0.902·21-s − 0.727·22-s − 1.20·23-s + 1.81·24-s + 1.17·25-s − 2.91·26-s − 0.192·27-s + 4.13·28-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
| \(L(1)\) |
\(\approx\) |
\(1.453293269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453293269\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 5.50T + 13T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 5.79T + 23T^{2} \) |
| 29 | \( 1 - 1.48T + 29T^{2} \) |
| 31 | \( 1 + 5.62T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + 6.58T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 1.45T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 0.751T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 8.70T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 - 4.33T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.85903891532213486400696511033, −7.48372035622166151751489137550, −6.39633286671902153398461366172, −6.01052663141266077830751365928, −5.52138295610277781549482919080, −4.40103685261736048336999231047, −3.09104640746465067579838363734, −1.90808557536013864476498179157, −1.59080587921593922480463285798, −0.916420513898560308875682877108,
0.916420513898560308875682877108, 1.59080587921593922480463285798, 1.90808557536013864476498179157, 3.09104640746465067579838363734, 4.40103685261736048336999231047, 5.52138295610277781549482919080, 6.01052663141266077830751365928, 6.39633286671902153398461366172, 7.48372035622166151751489137550, 7.85903891532213486400696511033
