L(s) = 1 | + 2.21·3-s − 3.59·5-s + 5.06·7-s + 1.90·9-s − 2.55·11-s + 4.93·13-s − 7.96·15-s + 2.68·17-s + 4.72·19-s + 11.2·21-s + 1.49·23-s + 7.93·25-s − 2.41·27-s − 2.43·29-s − 3.19·31-s − 5.67·33-s − 18.2·35-s − 2.90·37-s + 10.9·39-s − 41-s − 7.62·43-s − 6.86·45-s − 5.15·47-s + 18.6·49-s + 5.95·51-s + 11.1·53-s + 9.20·55-s + ⋯ |
L(s) = 1 | + 1.27·3-s − 1.60·5-s + 1.91·7-s + 0.636·9-s − 0.771·11-s + 1.36·13-s − 2.05·15-s + 0.651·17-s + 1.08·19-s + 2.44·21-s + 0.311·23-s + 1.58·25-s − 0.465·27-s − 0.451·29-s − 0.573·31-s − 0.987·33-s − 3.07·35-s − 0.478·37-s + 1.75·39-s − 0.156·41-s − 1.16·43-s − 1.02·45-s − 0.751·47-s + 2.66·49-s + 0.833·51-s + 1.52·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.803712009\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.803712009\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 + 3.59T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.28T + 73T^{2} \) |
| 79 | \( 1 - 4.28T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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
−8.422435620372730721706557429779, −8.186882690457284448756006624303, −7.72939453723444184419227731211, −7.07178675005068347233659777585, −5.45437959979957829225405970281, −4.85275643779867045779411188672, −3.70369039847120143227692159829, −3.47984219833737364464605768733, −2.17490939955186608283626330847, −1.06790017080150529400785535715,
1.06790017080150529400785535715, 2.17490939955186608283626330847, 3.47984219833737364464605768733, 3.70369039847120143227692159829, 4.85275643779867045779411188672, 5.45437959979957829225405970281, 7.07178675005068347233659777585, 7.72939453723444184419227731211, 8.186882690457284448756006624303, 8.422435620372730721706557429779