| L(s) = 1 | + 0.368·3-s + 2.18·5-s + 4.72·7-s − 2.86·9-s + 3.40·11-s − 4.21·13-s + 0.803·15-s − 1.82·17-s + 3.78·19-s + 1.73·21-s + 7.58·23-s − 0.235·25-s − 2.15·27-s − 8.32·29-s − 1.94·31-s + 1.25·33-s + 10.3·35-s + 8.92·37-s − 1.55·39-s − 1.04·41-s + 4.41·43-s − 6.25·45-s + 9.78·47-s + 15.2·49-s − 0.670·51-s + 11.7·53-s + 7.43·55-s + ⋯ |
| L(s) = 1 | + 0.212·3-s + 0.976·5-s + 1.78·7-s − 0.954·9-s + 1.02·11-s − 1.17·13-s + 0.207·15-s − 0.441·17-s + 0.868·19-s + 0.379·21-s + 1.58·23-s − 0.0471·25-s − 0.415·27-s − 1.54·29-s − 0.349·31-s + 0.218·33-s + 1.74·35-s + 1.46·37-s − 0.248·39-s − 0.162·41-s + 0.673·43-s − 0.932·45-s + 1.42·47-s + 2.18·49-s − 0.0938·51-s + 1.60·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.091968078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.091968078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.368T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 - 4.72T + 7T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 - 7.58T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 9.78T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 3.47T + 59T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 + 0.364T + 67T^{2} \) |
| 71 | \( 1 + 6.01T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 + 6.94T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 8.66T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
| 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
−8.596442753193865964977866296490, −7.56853577788637856459001857584, −7.25531064440376216664158950007, −6.02203941679942849801961799554, −5.43284787257254680816757090331, −4.84991656508719593572616556671, −3.93386691616069306272009137571, −2.65988153913616364108119173745, −2.03498912008287750811710527456, −1.06847065482922151706324607397,
1.06847065482922151706324607397, 2.03498912008287750811710527456, 2.65988153913616364108119173745, 3.93386691616069306272009137571, 4.84991656508719593572616556671, 5.43284787257254680816757090331, 6.02203941679942849801961799554, 7.25531064440376216664158950007, 7.56853577788637856459001857584, 8.596442753193865964977866296490
