| L(s) = 1 | − 2.65·2-s + 3-s + 5.03·4-s + 1.71·5-s − 2.65·6-s − 7-s − 8.04·8-s + 9-s − 4.54·10-s + 5.03·12-s + 1.51·13-s + 2.65·14-s + 1.71·15-s + 11.2·16-s − 2.39·17-s − 2.65·18-s + 6.33·19-s + 8.62·20-s − 21-s − 5.05·23-s − 8.04·24-s − 2.06·25-s − 4.00·26-s + 27-s − 5.03·28-s + 8.34·29-s − 4.54·30-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.577·3-s + 2.51·4-s + 0.766·5-s − 1.08·6-s − 0.377·7-s − 2.84·8-s + 0.333·9-s − 1.43·10-s + 1.45·12-s + 0.419·13-s + 0.708·14-s + 0.442·15-s + 2.81·16-s − 0.580·17-s − 0.625·18-s + 1.45·19-s + 1.92·20-s − 0.218·21-s − 1.05·23-s − 1.64·24-s − 0.413·25-s − 0.786·26-s + 0.192·27-s − 0.951·28-s + 1.54·29-s − 0.829·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082776910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082776910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 13 | \( 1 - 1.51T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 - 8.34T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 + 5.53T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 - 5.86T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.824738555241848757619674863751, −8.404858845048452112374498557121, −7.64983716212354805003231560418, −6.76508167573584883186224792967, −6.31333429803900662450536610164, −5.25755556620067484046913263082, −3.66174457620586926606403063253, −2.66760333215944080496524066908, −1.91647824789264311802059690484, −0.859212824965582697741151841781,
0.859212824965582697741151841781, 1.91647824789264311802059690484, 2.66760333215944080496524066908, 3.66174457620586926606403063253, 5.25755556620067484046913263082, 6.31333429803900662450536610164, 6.76508167573584883186224792967, 7.64983716212354805003231560418, 8.404858845048452112374498557121, 8.824738555241848757619674863751
