| L(s) = 1 | + 0.231·2-s − 3-s − 1.94·4-s − 0.231·6-s + 4.71·7-s − 0.913·8-s + 9-s − 3.55·11-s + 1.94·12-s − 3.00·13-s + 1.09·14-s + 3.68·16-s − 6.94·17-s + 0.231·18-s + 4.84·19-s − 4.71·21-s − 0.823·22-s + 3.68·23-s + 0.913·24-s − 0.696·26-s − 27-s − 9.18·28-s + 5.90·29-s − 3.00·31-s + 2.67·32-s + 3.55·33-s − 1.60·34-s + ⋯ |
| L(s) = 1 | + 0.163·2-s − 0.577·3-s − 0.973·4-s − 0.0944·6-s + 1.78·7-s − 0.322·8-s + 0.333·9-s − 1.07·11-s + 0.561·12-s − 0.834·13-s + 0.291·14-s + 0.920·16-s − 1.68·17-s + 0.0545·18-s + 1.11·19-s − 1.02·21-s − 0.175·22-s + 0.769·23-s + 0.186·24-s − 0.136·26-s − 0.192·27-s − 1.73·28-s + 1.09·29-s − 0.539·31-s + 0.473·32-s + 0.619·33-s − 0.275·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.253020389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253020389\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.231T + 2T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 + 3.00T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 - 9.98T + 37T^{2} \) |
| 41 | \( 1 + 5.71T + 41T^{2} \) |
| 43 | \( 1 - 0.0371T + 43T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 + 1.18T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 + 4.82T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 - 3.27T + 83T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−8.512358153667190505558917171340, −7.83344884421367329254038786838, −7.29624348611975633337682579952, −6.13899799505316731841498786430, −5.16697886950939905069739666359, −4.82624112089089297716296630639, −4.43242719880615351708950560500, −3.01363933175262424474553220735, −1.91716083683401017232295861719, −0.67275962926564992869686514572,
0.67275962926564992869686514572, 1.91716083683401017232295861719, 3.01363933175262424474553220735, 4.43242719880615351708950560500, 4.82624112089089297716296630639, 5.16697886950939905069739666359, 6.13899799505316731841498786430, 7.29624348611975633337682579952, 7.83344884421367329254038786838, 8.512358153667190505558917171340
