L(s) = 1 | + 2.51·3-s − 0.472·5-s + 4.16·7-s + 3.33·9-s − 0.304·11-s + 3.93·13-s − 1.18·15-s + 5.23·17-s − 3.05·19-s + 10.4·21-s + 7.82·23-s − 4.77·25-s + 0.847·27-s − 7.79·29-s − 9.12·31-s − 0.766·33-s − 1.96·35-s − 7.45·37-s + 9.90·39-s − 5.32·41-s + 10.2·43-s − 1.57·45-s + 8.20·47-s + 10.3·49-s + 13.1·51-s − 8.29·53-s + 0.143·55-s + ⋯ |
L(s) = 1 | + 1.45·3-s − 0.211·5-s + 1.57·7-s + 1.11·9-s − 0.0918·11-s + 1.09·13-s − 0.306·15-s + 1.26·17-s − 0.701·19-s + 2.28·21-s + 1.63·23-s − 0.955·25-s + 0.163·27-s − 1.44·29-s − 1.63·31-s − 0.133·33-s − 0.332·35-s − 1.22·37-s + 1.58·39-s − 0.832·41-s + 1.56·43-s − 0.234·45-s + 1.19·47-s + 1.48·49-s + 1.84·51-s − 1.13·53-s + 0.0193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.556432660\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.556432660\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 2.51T + 3T^{2} \) |
| 5 | \( 1 + 0.472T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 + 0.304T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 3.05T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 + 7.45T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 + 8.29T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 + 0.845T + 61T^{2} \) |
| 67 | \( 1 + 5.94T + 67T^{2} \) |
| 71 | \( 1 - 6.20T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 9.74T + 79T^{2} \) |
| 83 | \( 1 + 4.98T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 0.396T + 97T^{2} \) |
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
−8.817379041078856144775583208564, −8.539235587563849885741158603275, −7.54816456889488908190454564336, −7.42645578504342275940130719342, −5.83420110743480903754409063346, −5.06340094504036134278444406881, −3.91497878372608141029367459923, −3.42414511290684254817427013670, −2.14073974637787167363712516092, −1.39955481773998976069350032439,
1.39955481773998976069350032439, 2.14073974637787167363712516092, 3.42414511290684254817427013670, 3.91497878372608141029367459923, 5.06340094504036134278444406881, 5.83420110743480903754409063346, 7.42645578504342275940130719342, 7.54816456889488908190454564336, 8.539235587563849885741158603275, 8.817379041078856144775583208564