L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 0.487·7-s + 8-s + 9-s + 10-s − 5.20·11-s + 12-s + 4.12·13-s + 0.487·14-s + 15-s + 16-s + 3.67·17-s + 18-s + 0.430·19-s + 20-s + 0.487·21-s − 5.20·22-s + 7.31·23-s + 24-s + 25-s + 4.12·26-s + 27-s + 0.487·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.184·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.57·11-s + 0.288·12-s + 1.14·13-s + 0.130·14-s + 0.258·15-s + 0.250·16-s + 0.891·17-s + 0.235·18-s + 0.0986·19-s + 0.223·20-s + 0.106·21-s − 1.11·22-s + 1.52·23-s + 0.204·24-s + 0.200·25-s + 0.808·26-s + 0.192·27-s + 0.0921·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.385529281\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.385529281\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 0.487T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 - 3.67T + 17T^{2} \) |
| 19 | \( 1 - 0.430T + 19T^{2} \) |
| 23 | \( 1 - 7.31T + 23T^{2} \) |
| 29 | \( 1 + 6.77T + 29T^{2} \) |
| 31 | \( 1 - 7.80T + 31T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 3.52T + 47T^{2} \) |
| 53 | \( 1 + 7.67T + 53T^{2} \) |
| 59 | \( 1 + 0.974T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 1.19T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 + 7.15T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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
−9.965134280770507071939025302068, −8.959573761759463273629336769266, −8.083090202187354978017654106298, −7.42965784754398170579912385024, −6.34431787794355415468557712204, −5.44978487914199118205939634699, −4.74750980364201892268137178643, −3.42752121929457007523382079861, −2.75847261528225599328467992197, −1.46119511378691107243432288840,
1.46119511378691107243432288840, 2.75847261528225599328467992197, 3.42752121929457007523382079861, 4.74750980364201892268137178643, 5.44978487914199118205939634699, 6.34431787794355415468557712204, 7.42965784754398170579912385024, 8.083090202187354978017654106298, 8.959573761759463273629336769266, 9.965134280770507071939025302068