| L(s) = 1 | − 6·5-s + 49·7-s + 108·11-s − 346·13-s + 1.39e3·17-s − 1.01e3·19-s + 1.53e3·23-s − 3.08e3·25-s + 3.76e3·29-s − 736·31-s − 294·35-s + 2.05e3·37-s + 1.55e4·41-s + 1.10e4·43-s − 4.56e3·47-s + 2.40e3·49-s + 7.96e3·53-s − 648·55-s + 7.02e3·59-s + 2.68e4·61-s + 2.07e3·65-s + 5.21e4·67-s + 2.54e3·71-s − 9.76e3·73-s + 5.29e3·77-s + 6.86e4·79-s + 6.16e4·83-s + ⋯ |
| L(s) = 1 | − 0.107·5-s + 0.377·7-s + 0.269·11-s − 0.567·13-s + 1.17·17-s − 0.643·19-s + 0.605·23-s − 0.988·25-s + 0.830·29-s − 0.137·31-s − 0.0405·35-s + 0.246·37-s + 1.44·41-s + 0.910·43-s − 0.301·47-s + 1/7·49-s + 0.389·53-s − 0.0288·55-s + 0.262·59-s + 0.924·61-s + 0.0609·65-s + 1.41·67-s + 0.0598·71-s − 0.214·73-s + 0.101·77-s + 1.23·79-s + 0.982·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.065885982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.065885982\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 6 T + p^{5} T^{2} \) |
| 11 | \( 1 - 108 T + p^{5} T^{2} \) |
| 13 | \( 1 + 346 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1398 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1012 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1536 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3762 T + p^{5} T^{2} \) |
| 31 | \( 1 + 736 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2054 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15534 T + p^{5} T^{2} \) |
| 43 | \( 1 - 11036 T + p^{5} T^{2} \) |
| 47 | \( 1 + 4560 T + p^{5} T^{2} \) |
| 53 | \( 1 - 7962 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7020 T + p^{5} T^{2} \) |
| 61 | \( 1 - 26870 T + p^{5} T^{2} \) |
| 67 | \( 1 - 52148 T + p^{5} T^{2} \) |
| 71 | \( 1 - 2544 T + p^{5} T^{2} \) |
| 73 | \( 1 + 9766 T + p^{5} T^{2} \) |
| 79 | \( 1 - 68672 T + p^{5} T^{2} \) |
| 83 | \( 1 - 61668 T + p^{5} T^{2} \) |
| 89 | \( 1 - 41454 T + p^{5} T^{2} \) |
| 97 | \( 1 + 111262 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23138089357119679135756667283, −10.21752015582178436864486328537, −9.305923930445000485522421110441, −8.170712503255528215081055833433, −7.33645517829191876582811188243, −6.09874442843746341317367875863, −4.97505711304533382875399774759, −3.79741732738701173555625412285, −2.36176606138213056498122775662, −0.860910255202223071753380982438,
0.860910255202223071753380982438, 2.36176606138213056498122775662, 3.79741732738701173555625412285, 4.97505711304533382875399774759, 6.09874442843746341317367875863, 7.33645517829191876582811188243, 8.170712503255528215081055833433, 9.305923930445000485522421110441, 10.21752015582178436864486328537, 11.23138089357119679135756667283
