| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 4.81·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 9.63·10-s + 43.0·11-s − 12·12-s + 48.6·13-s + 14·14-s − 14.4·15-s + 16·16-s − 24.7·17-s + 18·18-s + 63.2·19-s + 19.2·20-s − 21·21-s + 86.0·22-s − 23·23-s − 24·24-s − 101.·25-s + 97.3·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.431·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.304·10-s + 1.17·11-s − 0.288·12-s + 1.03·13-s + 0.267·14-s − 0.248·15-s + 0.250·16-s − 0.353·17-s + 0.235·18-s + 0.763·19-s + 0.215·20-s − 0.218·21-s + 0.834·22-s − 0.208·23-s − 0.204·24-s − 0.814·25-s + 0.734·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.668826804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.668826804\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 4.81T + 125T^{2} \) |
| 11 | \( 1 - 43.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 63.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 4.70T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 465.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 513.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 428.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 158.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 28.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 797.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 778.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 526.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 108.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 409.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 888.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 255.T + 9.12e5T^{2} \) |
| 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
−9.688798591399700340350621645265, −8.907455920075984879001147103697, −7.77986730190477973009711503547, −6.79993553985949136435972669349, −6.06603659364882314503565473841, −5.40972937367743305560591128242, −4.29702838143686037608451591320, −3.55719272965556093408953065427, −2.01949348780647044692893375241, −1.03078390485532278505089932112,
1.03078390485532278505089932112, 2.01949348780647044692893375241, 3.55719272965556093408953065427, 4.29702838143686037608451591320, 5.40972937367743305560591128242, 6.06603659364882314503565473841, 6.79993553985949136435972669349, 7.77986730190477973009711503547, 8.907455920075984879001147103697, 9.688798591399700340350621645265
