| L(s) = 1 | + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s − 3·11-s − 4·13-s + 14-s + 16-s + 6·17-s − 7·19-s + 3·20-s − 3·22-s + 3·23-s + 4·25-s − 4·26-s + 28-s + 5·31-s + 32-s + 6·34-s + 3·35-s − 7·37-s − 7·38-s + 3·40-s + 9·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s − 0.904·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.60·19-s + 0.670·20-s − 0.639·22-s + 0.625·23-s + 4/5·25-s − 0.784·26-s + 0.188·28-s + 0.898·31-s + 0.176·32-s + 1.02·34-s + 0.507·35-s − 1.15·37-s − 1.13·38-s + 0.474·40-s + 1.40·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.398893991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.398893991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−11.42061259594256530369121590340, −10.26708504332289108173004023441, −9.938898618957151166860023695517, −8.539023583449884225816408011720, −7.46960982382182631181281303264, −6.34863993594311576079450533659, −5.42330356244252482781728099229, −4.70886712751954702153319801692, −2.95503675125733963986543239116, −1.90035837414507085552956582982,
1.90035837414507085552956582982, 2.95503675125733963986543239116, 4.70886712751954702153319801692, 5.42330356244252482781728099229, 6.34863993594311576079450533659, 7.46960982382182631181281303264, 8.539023583449884225816408011720, 9.938898618957151166860023695517, 10.26708504332289108173004023441, 11.42061259594256530369121590340
