| L(s) = 1 | + 2.78·2-s + 0.869·3-s + 5.73·4-s − 5-s + 2.41·6-s + 7-s + 10.3·8-s − 2.24·9-s − 2.78·10-s − 3.60·11-s + 4.98·12-s − 4.41·13-s + 2.78·14-s − 0.869·15-s + 17.4·16-s − 17-s − 6.24·18-s − 0.0409·19-s − 5.73·20-s + 0.869·21-s − 10.0·22-s + 5.69·23-s + 9.02·24-s + 25-s − 12.2·26-s − 4.55·27-s + 5.73·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.501·3-s + 2.86·4-s − 0.447·5-s + 0.986·6-s + 0.377·7-s + 3.67·8-s − 0.748·9-s − 0.879·10-s − 1.08·11-s + 1.43·12-s − 1.22·13-s + 0.743·14-s − 0.224·15-s + 4.35·16-s − 0.242·17-s − 1.47·18-s − 0.00939·19-s − 1.28·20-s + 0.189·21-s − 2.13·22-s + 1.18·23-s + 1.84·24-s + 0.200·25-s − 2.40·26-s − 0.877·27-s + 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.852095358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.852095358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 3 | \( 1 - 0.869T + 3T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 19 | \( 1 + 0.0409T + 19T^{2} \) |
| 23 | \( 1 - 5.69T + 23T^{2} \) |
| 29 | \( 1 + 2.07T + 29T^{2} \) |
| 31 | \( 1 + 4.35T + 31T^{2} \) |
| 37 | \( 1 + 6.59T + 37T^{2} \) |
| 41 | \( 1 + 0.416T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 - 7.54T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 + 0.995T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 6.06T + 97T^{2} \) |
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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
−11.09799268700733679688886516224, −10.24167974647445759678603521222, −8.677441098037366025341352499051, −7.53295713230238771572361544695, −7.13185894333766576908140913292, −5.63274341309521433730396217590, −5.12226920286361295124205582422, −4.08614422688513936999119869048, −2.96834033959285961382845744688, −2.28832452360743484191079679215,
2.28832452360743484191079679215, 2.96834033959285961382845744688, 4.08614422688513936999119869048, 5.12226920286361295124205582422, 5.63274341309521433730396217590, 7.13185894333766576908140913292, 7.53295713230238771572361544695, 8.677441098037366025341352499051, 10.24167974647445759678603521222, 11.09799268700733679688886516224
