| L(s) = 1 | + 0.972·2-s − 3-s − 1.05·4-s + 4.14·5-s − 0.972·6-s − 7-s − 2.97·8-s + 9-s + 4.02·10-s + 0.996·11-s + 1.05·12-s + 0.889·13-s − 0.972·14-s − 4.14·15-s − 0.783·16-s + 4.67·17-s + 0.972·18-s + 2.62·19-s − 4.36·20-s + 21-s + 0.970·22-s + 5.36·23-s + 2.97·24-s + 12.1·25-s + 0.865·26-s − 27-s + 1.05·28-s + ⋯ |
| L(s) = 1 | + 0.688·2-s − 0.577·3-s − 0.526·4-s + 1.85·5-s − 0.397·6-s − 0.377·7-s − 1.05·8-s + 0.333·9-s + 1.27·10-s + 0.300·11-s + 0.304·12-s + 0.246·13-s − 0.260·14-s − 1.06·15-s − 0.195·16-s + 1.13·17-s + 0.229·18-s + 0.603·19-s − 0.975·20-s + 0.218·21-s + 0.206·22-s + 1.11·23-s + 0.606·24-s + 2.43·25-s + 0.169·26-s − 0.192·27-s + 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.686729705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.686729705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 0.972T + 2T^{2} \) |
| 5 | \( 1 - 4.14T + 5T^{2} \) |
| 11 | \( 1 - 0.996T + 11T^{2} \) |
| 13 | \( 1 - 0.889T + 13T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 - 5.36T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 7.58T + 31T^{2} \) |
| 37 | \( 1 - 0.373T + 37T^{2} \) |
| 41 | \( 1 + 2.28T + 41T^{2} \) |
| 43 | \( 1 + 7.58T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 0.612T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 6.43T + 79T^{2} \) |
| 83 | \( 1 + 2.88T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−8.828823426302757221963126024947, −7.45439591105059884528438644080, −6.68772913318529408121444967554, −5.89846287007377595165457578685, −5.44445208361376034723966790413, −5.10420277407267335846041987789, −3.83175065650674633365942546618, −3.13774078555470410646848629894, −1.96702497386314623732645470739, −0.914594548635652655346586395679,
0.914594548635652655346586395679, 1.96702497386314623732645470739, 3.13774078555470410646848629894, 3.83175065650674633365942546618, 5.10420277407267335846041987789, 5.44445208361376034723966790413, 5.89846287007377595165457578685, 6.68772913318529408121444967554, 7.45439591105059884528438644080, 8.828823426302757221963126024947
