| L(s) = 1 | + 2·2-s + 4·4-s + 14·5-s − 7·7-s + 8·8-s + 28·10-s − 11·11-s + 38·13-s − 14·14-s + 16·16-s − 54·17-s + 40·19-s + 56·20-s − 22·22-s − 8·23-s + 71·25-s + 76·26-s − 28·28-s + 170·29-s + 92·31-s + 32·32-s − 108·34-s − 98·35-s + 294·37-s + 80·38-s + 112·40-s + 258·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.25·5-s − 0.377·7-s + 0.353·8-s + 0.885·10-s − 0.301·11-s + 0.810·13-s − 0.267·14-s + 1/4·16-s − 0.770·17-s + 0.482·19-s + 0.626·20-s − 0.213·22-s − 0.0725·23-s + 0.567·25-s + 0.573·26-s − 0.188·28-s + 1.08·29-s + 0.533·31-s + 0.176·32-s − 0.544·34-s − 0.473·35-s + 1.30·37-s + 0.341·38-s + 0.442·40-s + 0.982·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.591005222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.591005222\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 T + p^{3} T^{2} \) |
| 29 | \( 1 - 170 T + p^{3} T^{2} \) |
| 31 | \( 1 - 92 T + p^{3} T^{2} \) |
| 37 | \( 1 - 294 T + p^{3} T^{2} \) |
| 41 | \( 1 - 258 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 76 T + p^{3} T^{2} \) |
| 53 | \( 1 - 322 T + p^{3} T^{2} \) |
| 59 | \( 1 + 260 T + p^{3} T^{2} \) |
| 61 | \( 1 - 22 T + p^{3} T^{2} \) |
| 67 | \( 1 + 436 T + p^{3} T^{2} \) |
| 71 | \( 1 - 368 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 T + p^{3} T^{2} \) |
| 79 | \( 1 + 200 T + p^{3} T^{2} \) |
| 83 | \( 1 - 952 T + p^{3} T^{2} \) |
| 89 | \( 1 - 70 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 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
−9.324245631410623736347659563994, −8.459931870195348166878874576035, −7.39975318403041806765718851409, −6.33096089891552510457493782868, −6.04983059237703409190300319664, −5.07585906373371268378362870029, −4.15285901027199055127791576529, −2.97195014472816331651961715846, −2.19577459424050434072290453825, −0.995473766092741112284988907525,
0.995473766092741112284988907525, 2.19577459424050434072290453825, 2.97195014472816331651961715846, 4.15285901027199055127791576529, 5.07585906373371268378362870029, 6.04983059237703409190300319664, 6.33096089891552510457493782868, 7.39975318403041806765718851409, 8.459931870195348166878874576035, 9.324245631410623736347659563994
