| L(s) = 1 | + 2.74·2-s + 3-s + 5.56·4-s + 2.09·5-s + 2.74·6-s − 7-s + 9.79·8-s + 9-s + 5.74·10-s − 4.71·11-s + 5.56·12-s − 4.68·13-s − 2.74·14-s + 2.09·15-s + 15.8·16-s + 6.73·17-s + 2.74·18-s + 6.34·19-s + 11.6·20-s − 21-s − 12.9·22-s + 4.65·23-s + 9.79·24-s − 0.629·25-s − 12.8·26-s + 27-s − 5.56·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.78·4-s + 0.934·5-s + 1.12·6-s − 0.377·7-s + 3.46·8-s + 0.333·9-s + 1.81·10-s − 1.42·11-s + 1.60·12-s − 1.29·13-s − 0.734·14-s + 0.539·15-s + 3.95·16-s + 1.63·17-s + 0.648·18-s + 1.45·19-s + 2.60·20-s − 0.218·21-s − 2.76·22-s + 0.970·23-s + 1.99·24-s − 0.125·25-s − 2.52·26-s + 0.192·27-s − 1.05·28-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
| \(L(1)\) |
\(\approx\) |
\(9.528423537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.528423537\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.74T + 2T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 + 0.500T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 0.899T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 9.28T + 53T^{2} \) |
| 59 | \( 1 + 1.97T + 59T^{2} \) |
| 61 | \( 1 - 2.34T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 1.33T + 71T^{2} \) |
| 73 | \( 1 - 8.40T + 73T^{2} \) |
| 79 | \( 1 + 9.74T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 4.55T + 97T^{2} \) |
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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
−7.998496042981781413982814797810, −7.38151662827541139983234138539, −6.98849234598962829970641383485, −5.75506781826841889670060709514, −5.35857933064723301676440800053, −4.97424846845060479536995148502, −3.68494706403489171984244809325, −3.01675751215870703484255030235, −2.51068749709184900106448375469, −1.54641863201784365266064128573,
1.54641863201784365266064128573, 2.51068749709184900106448375469, 3.01675751215870703484255030235, 3.68494706403489171984244809325, 4.97424846845060479536995148502, 5.35857933064723301676440800053, 5.75506781826841889670060709514, 6.98849234598962829970641383485, 7.38151662827541139983234138539, 7.998496042981781413982814797810
