| L(s) = 1 | − 2.67·2-s + 5.13·4-s − 2.04·7-s − 8.37·8-s − 3.94·11-s − 5.35·13-s + 5.47·14-s + 12.1·16-s − 6.45·17-s − 19-s + 10.5·22-s − 3.48·23-s + 14.2·26-s − 10.5·28-s − 8.34·29-s − 2.96·31-s − 15.5·32-s + 17.2·34-s + 6.42·37-s + 2.67·38-s − 11.4·41-s − 7.22·43-s − 20.2·44-s + 9.30·46-s + 1.11·47-s − 2.79·49-s − 27.4·52-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 2.56·4-s − 0.774·7-s − 2.96·8-s − 1.18·11-s − 1.48·13-s + 1.46·14-s + 3.02·16-s − 1.56·17-s − 0.229·19-s + 2.24·22-s − 0.726·23-s + 2.80·26-s − 1.98·28-s − 1.55·29-s − 0.532·31-s − 2.75·32-s + 2.95·34-s + 1.05·37-s + 0.433·38-s − 1.78·41-s − 1.10·43-s − 3.05·44-s + 1.37·46-s + 0.162·47-s − 0.399·49-s − 3.81·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02795232421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02795232421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 8.34T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 6.42T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 7.22T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 - 0.362T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 + 0.135T + 61T^{2} \) |
| 67 | \( 1 + 5.17T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 7.20T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 + 5.35T + 97T^{2} \) |
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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
−8.432490395050098033199937729373, −7.76925968864850469521981510891, −7.15779045896750336697368939049, −6.60457459817082461808340051882, −5.75729933135082041557491878682, −4.76653820899643663812279423069, −3.37796121268832989332934211187, −2.41368588601645207062854269688, −1.95549679958958563332273104671, −0.11400901706099152037913327283,
0.11400901706099152037913327283, 1.95549679958958563332273104671, 2.41368588601645207062854269688, 3.37796121268832989332934211187, 4.76653820899643663812279423069, 5.75729933135082041557491878682, 6.60457459817082461808340051882, 7.15779045896750336697368939049, 7.76925968864850469521981510891, 8.432490395050098033199937729373
