L(s) = 1 | + 1.61·3-s + 3.85·5-s + 3.23·7-s − 0.381·9-s + 1.38·11-s − 2.85·13-s + 6.23·15-s − 4.47·17-s − 4.47·19-s + 5.23·21-s − 2.85·23-s + 9.85·25-s − 5.47·27-s − 9.32·29-s − 7.38·31-s + 2.23·33-s + 12.4·35-s − 37-s − 4.61·39-s + 9.61·41-s + 5.23·43-s − 1.47·45-s + 1.23·47-s + 3.47·49-s − 7.23·51-s + 0.472·53-s + 5.32·55-s + ⋯ |
L(s) = 1 | + 0.934·3-s + 1.72·5-s + 1.22·7-s − 0.127·9-s + 0.416·11-s − 0.791·13-s + 1.61·15-s − 1.08·17-s − 1.02·19-s + 1.14·21-s − 0.595·23-s + 1.97·25-s − 1.05·27-s − 1.73·29-s − 1.32·31-s + 0.389·33-s + 2.10·35-s − 0.164·37-s − 0.739·39-s + 1.50·41-s + 0.798·43-s − 0.219·45-s + 0.180·47-s + 0.496·49-s − 1.01·51-s + 0.0648·53-s + 0.718·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.617798977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.617798977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 9.32T + 29T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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
−10.67727694313713790864342251250, −9.449396245340141849732893883610, −9.135860323182370177548760714184, −8.218471364592455277411805112025, −7.20426838977333196831701412803, −6.02777335083143123460013540802, −5.22279688613488936545507250482, −4.03188932570837349324041059386, −2.25995673408629056447611980325, −1.99810976681813550970699973392,
1.99810976681813550970699973392, 2.25995673408629056447611980325, 4.03188932570837349324041059386, 5.22279688613488936545507250482, 6.02777335083143123460013540802, 7.20426838977333196831701412803, 8.218471364592455277411805112025, 9.135860323182370177548760714184, 9.449396245340141849732893883610, 10.67727694313713790864342251250