| L(s) = 1 | + 4·2-s + 29·3-s + 16·4-s + 31·5-s + 116·6-s + 64·8-s + 598·9-s + 124·10-s + 121·11-s + 464·12-s − 112·13-s + 899·15-s + 256·16-s + 1.14e3·17-s + 2.39e3·18-s + 612·19-s + 496·20-s + 484·22-s − 1.94e3·23-s + 1.85e3·24-s − 2.16e3·25-s − 448·26-s + 1.02e4·27-s + 1.19e3·29-s + 3.59e3·30-s + 1.03e3·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.86·3-s + 1/2·4-s + 0.554·5-s + 1.31·6-s + 0.353·8-s + 2.46·9-s + 0.392·10-s + 0.301·11-s + 0.930·12-s − 0.183·13-s + 1.03·15-s + 1/4·16-s + 0.958·17-s + 1.74·18-s + 0.388·19-s + 0.277·20-s + 0.213·22-s − 0.765·23-s + 0.657·24-s − 0.692·25-s − 0.129·26-s + 2.71·27-s + 0.263·29-s + 0.729·30-s + 0.193·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.962486859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.962486859\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 29 T + p^{5} T^{2} \) |
| 5 | \( 1 - 31 T + p^{5} T^{2} \) |
| 13 | \( 1 + 112 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1142 T + p^{5} T^{2} \) |
| 19 | \( 1 - 612 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1941 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1192 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1037 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8083 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10444 T + p^{5} T^{2} \) |
| 43 | \( 1 - 58 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8656 T + p^{5} T^{2} \) |
| 53 | \( 1 + 20318 T + p^{5} T^{2} \) |
| 59 | \( 1 - 21351 T + p^{5} T^{2} \) |
| 61 | \( 1 + 47044 T + p^{5} T^{2} \) |
| 67 | \( 1 - 48093 T + p^{5} T^{2} \) |
| 71 | \( 1 + 24967 T + p^{5} T^{2} \) |
| 73 | \( 1 - 42288 T + p^{5} T^{2} \) |
| 79 | \( 1 + 72410 T + p^{5} T^{2} \) |
| 83 | \( 1 - 15806 T + p^{5} T^{2} \) |
| 89 | \( 1 - 114761 T + p^{5} T^{2} \) |
| 97 | \( 1 - 5159 T + p^{5} T^{2} \) |
| 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
−9.328898202158113156426938925339, −8.098730830949982840043753109244, −7.73892329173357746231889590542, −6.70717820675724747700699352561, −5.73203850560298424301764656981, −4.53546085862631976772751973829, −3.71958995453164891625259487045, −2.92066619618526002192670093929, −2.11922484778386016175477453926, −1.23648988697166518296442429209,
1.23648988697166518296442429209, 2.11922484778386016175477453926, 2.92066619618526002192670093929, 3.71958995453164891625259487045, 4.53546085862631976772751973829, 5.73203850560298424301764656981, 6.70717820675724747700699352561, 7.73892329173357746231889590542, 8.098730830949982840043753109244, 9.328898202158113156426938925339
