L(s) = 1 | + 0.416·2-s + 1.40·3-s − 1.82·4-s − 5-s + 0.584·6-s − 1.52·7-s − 1.59·8-s − 1.03·9-s − 0.416·10-s + 11-s − 2.56·12-s + 2.05·13-s − 0.637·14-s − 1.40·15-s + 2.98·16-s + 6.91·17-s − 0.430·18-s − 1.02·19-s + 1.82·20-s − 2.14·21-s + 0.416·22-s − 0.555·23-s − 2.23·24-s + 25-s + 0.858·26-s − 5.65·27-s + 2.79·28-s + ⋯ |
L(s) = 1 | + 0.294·2-s + 0.809·3-s − 0.913·4-s − 0.447·5-s + 0.238·6-s − 0.577·7-s − 0.563·8-s − 0.344·9-s − 0.131·10-s + 0.301·11-s − 0.739·12-s + 0.571·13-s − 0.170·14-s − 0.362·15-s + 0.747·16-s + 1.67·17-s − 0.101·18-s − 0.235·19-s + 0.408·20-s − 0.468·21-s + 0.0888·22-s − 0.115·23-s − 0.456·24-s + 0.200·25-s + 0.168·26-s − 1.08·27-s + 0.527·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 - 0.416T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 13 | \( 1 - 2.05T + 13T^{2} \) |
| 17 | \( 1 - 6.91T + 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 + 0.555T + 23T^{2} \) |
| 29 | \( 1 - 8.69T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 7.10T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 3.95T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 3.25T + 71T^{2} \) |
| 79 | \( 1 + 8.38T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 6.12T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.113120181494657202415807736591, −7.64136331426232090437547194929, −6.53827620772107732819069751294, −5.75610699305362585298635539614, −5.05181073360994030934947811341, −3.99087688824791355907845857784, −3.43405542757590417138194564453, −2.93145006291258470646122824306, −1.38819387810353001661151637366, 0,
1.38819387810353001661151637366, 2.93145006291258470646122824306, 3.43405542757590417138194564453, 3.99087688824791355907845857784, 5.05181073360994030934947811341, 5.75610699305362585298635539614, 6.53827620772107732819069751294, 7.64136331426232090437547194929, 8.113120181494657202415807736591