| L(s) = 1 | − 2-s − 3-s + 4-s + 3.59·5-s + 6-s + 1.23·7-s − 8-s + 9-s − 3.59·10-s + 0.260·11-s − 12-s + 5.91·13-s − 1.23·14-s − 3.59·15-s + 16-s − 4.05·17-s − 18-s + 6.94·19-s + 3.59·20-s − 1.23·21-s − 0.260·22-s + 23-s + 24-s + 7.94·25-s − 5.91·26-s − 27-s + 1.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.60·5-s + 0.408·6-s + 0.464·7-s − 0.353·8-s + 0.333·9-s − 1.13·10-s + 0.0784·11-s − 0.288·12-s + 1.64·13-s − 0.328·14-s − 0.929·15-s + 0.250·16-s − 0.984·17-s − 0.235·18-s + 1.59·19-s + 0.804·20-s − 0.268·21-s − 0.0554·22-s + 0.208·23-s + 0.204·24-s + 1.58·25-s − 1.16·26-s − 0.192·27-s + 0.232·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.953065197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.953065197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 0.260T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 + 4.05T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 1.49T + 41T^{2} \) |
| 43 | \( 1 + 2.42T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 + 2.88T + 53T^{2} \) |
| 59 | \( 1 - 0.426T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 - 5.02T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
| 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
−8.546230897692057138533108236243, −7.83330739329757817726815651296, −6.70340990599368591092550559535, −6.39604839453110415759367122178, −5.56173003943525570699708667975, −5.05008325526529854422952873931, −3.79117649733069022940862403101, −2.63980689308710913991800128661, −1.64627094991400125478324745277, −1.02653446190447595663648990950,
1.02653446190447595663648990950, 1.64627094991400125478324745277, 2.63980689308710913991800128661, 3.79117649733069022940862403101, 5.05008325526529854422952873931, 5.56173003943525570699708667975, 6.39604839453110415759367122178, 6.70340990599368591092550559535, 7.83330739329757817726815651296, 8.546230897692057138533108236243
