L(s) = 1 | − 2-s + 4-s + 3.35·7-s − 8-s + 0.962·11-s + 1.61·13-s − 3.35·14-s + 16-s − 0.387·17-s + 19-s − 0.962·22-s − 0.962·23-s − 1.61·26-s + 3.35·28-s − 6.96·29-s + 3.35·31-s − 32-s + 0.387·34-s + 1.61·37-s − 38-s + 9.27·41-s − 6.18·43-s + 0.962·44-s + 0.962·46-s + 0.962·47-s + 4.22·49-s + 1.61·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.26·7-s − 0.353·8-s + 0.290·11-s + 0.447·13-s − 0.895·14-s + 0.250·16-s − 0.0940·17-s + 0.229·19-s − 0.205·22-s − 0.200·23-s − 0.316·26-s + 0.633·28-s − 1.29·29-s + 0.601·31-s − 0.176·32-s + 0.0665·34-s + 0.265·37-s − 0.162·38-s + 1.44·41-s − 0.943·43-s + 0.145·44-s + 0.141·46-s + 0.140·47-s + 0.603·49-s + 0.223·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.853692145\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.853692145\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 - 0.962T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 0.387T + 17T^{2} \) |
| 23 | \( 1 + 0.962T + 23T^{2} \) |
| 29 | \( 1 + 6.96T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 - 9.27T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 - 0.962T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.22T + 67T^{2} \) |
| 71 | \( 1 + 7.22T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 + 3.35T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 4.64T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.80018830202687704003663921511, −7.35849836485321512028946442656, −6.47324647198573705915022576547, −5.79047488542164358393696223561, −5.04839638533060583006422556797, −4.25368753709149120197544486505, −3.46286787526812040789781795366, −2.35683643481579403163961841076, −1.65570175032395001083512406353, −0.77727258028420918011490586216,
0.77727258028420918011490586216, 1.65570175032395001083512406353, 2.35683643481579403163961841076, 3.46286787526812040789781795366, 4.25368753709149120197544486505, 5.04839638533060583006422556797, 5.79047488542164358393696223561, 6.47324647198573705915022576547, 7.35849836485321512028946442656, 7.80018830202687704003663921511