L(s) = 1 | + 3.23·5-s − 7-s − 3.23·11-s − 4.47·13-s − 0.763·17-s + 2.47·19-s − 5.70·23-s + 5.47·25-s + 1.52·29-s + 2.47·31-s − 3.23·35-s + 4.47·37-s − 7.23·41-s − 12.9·43-s − 1.52·47-s + 49-s + 8·53-s − 10.4·55-s − 6.47·59-s + 4.47·61-s − 14.4·65-s − 8·67-s − 7.23·71-s − 14.9·73-s + 3.23·77-s − 12.9·79-s + 12.9·83-s + ⋯ |
L(s) = 1 | + 1.44·5-s − 0.377·7-s − 0.975·11-s − 1.24·13-s − 0.185·17-s + 0.567·19-s − 1.19·23-s + 1.09·25-s + 0.283·29-s + 0.444·31-s − 0.546·35-s + 0.735·37-s − 1.13·41-s − 1.97·43-s − 0.222·47-s + 0.142·49-s + 1.09·53-s − 1.41·55-s − 0.842·59-s + 0.572·61-s − 1.79·65-s − 0.977·67-s − 0.858·71-s − 1.74·73-s + 0.368·77-s − 1.45·79-s + 1.42·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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
−8.080247962880524239992727602077, −7.28654764573895619810892890771, −6.53623682500073696475950332695, −5.79168385884667625761192220370, −5.21676319628099153753763485514, −4.47094917554259473781801761593, −3.11164234038604366807686432136, −2.46109911371293024007085139854, −1.63263565704737372325808870495, 0,
1.63263565704737372325808870495, 2.46109911371293024007085139854, 3.11164234038604366807686432136, 4.47094917554259473781801761593, 5.21676319628099153753763485514, 5.79168385884667625761192220370, 6.53623682500073696475950332695, 7.28654764573895619810892890771, 8.080247962880524239992727602077