| L(s) = 1 | + 0.544·2-s − 3-s − 1.70·4-s + 2.42·5-s − 0.544·6-s + 7-s − 2.01·8-s + 9-s + 1.32·10-s + 4.74·11-s + 1.70·12-s + 6.42·13-s + 0.544·14-s − 2.42·15-s + 2.30·16-s + 7.11·17-s + 0.544·18-s + 0.432·19-s − 4.13·20-s − 21-s + 2.58·22-s − 4.15·23-s + 2.01·24-s + 0.900·25-s + 3.49·26-s − 27-s − 1.70·28-s + ⋯ |
| L(s) = 1 | + 0.385·2-s − 0.577·3-s − 0.851·4-s + 1.08·5-s − 0.222·6-s + 0.377·7-s − 0.712·8-s + 0.333·9-s + 0.418·10-s + 1.43·11-s + 0.491·12-s + 1.78·13-s + 0.145·14-s − 0.627·15-s + 0.577·16-s + 1.72·17-s + 0.128·18-s + 0.0992·19-s − 0.925·20-s − 0.218·21-s + 0.550·22-s − 0.866·23-s + 0.411·24-s + 0.180·25-s + 0.685·26-s − 0.192·27-s − 0.321·28-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
| \(L(1)\) |
\(\approx\) |
\(2.567319304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.567319304\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.544T + 2T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 - 7.11T + 17T^{2} \) |
| 19 | \( 1 - 0.432T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 - 7.53T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 5.63T + 37T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 + 8.34T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 - 0.129T + 61T^{2} \) |
| 67 | \( 1 - 0.621T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 5.75T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.496610145762239176706301445964, −7.88673935224945352972550245767, −6.41890322128573827009053322567, −6.23906916437665100410082136977, −5.51788165470709264736864773688, −4.79849629567085438112635756753, −3.86299724258185895431245953113, −3.30431898411474312838116149169, −1.60800734003683571442258736054, −1.04433168605812924796635375254,
1.04433168605812924796635375254, 1.60800734003683571442258736054, 3.30431898411474312838116149169, 3.86299724258185895431245953113, 4.79849629567085438112635756753, 5.51788165470709264736864773688, 6.23906916437665100410082136977, 6.41890322128573827009053322567, 7.88673935224945352972550245767, 8.496610145762239176706301445964
