| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 2·12-s − 4·13-s − 14-s + 2·15-s + 16-s + 18-s − 4·19-s + 20-s − 2·21-s + 22-s + 2·24-s + 25-s − 4·26-s − 4·27-s − 28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.577·12-s − 1.10·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.436·21-s + 0.213·22-s + 0.408·24-s + 1/5·25-s − 0.784·26-s − 0.769·27-s − 0.188·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.571146087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.571146087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.12593603431703, −12.48224395814204, −12.25840456186237, −11.73627361489813, −11.15982605459676, −10.45829290067280, −10.19229649949594, −9.667855283636740, −9.187451630741807, −8.731405314921411, −8.289768164018683, −7.638496858764075, −7.372947332049898, −6.647396023697873, −6.158710643781553, −5.946034795802521, −4.928294883655353, −4.713428552580991, −4.115990228968891, −3.517883186825322, −2.903302593654912, −2.540656025113869, −2.183836625091302, −1.395113322815686, −0.5500542787844422,
0.5500542787844422, 1.395113322815686, 2.183836625091302, 2.540656025113869, 2.903302593654912, 3.517883186825322, 4.115990228968891, 4.713428552580991, 4.928294883655353, 5.946034795802521, 6.158710643781553, 6.647396023697873, 7.372947332049898, 7.638496858764075, 8.289768164018683, 8.731405314921411, 9.187451630741807, 9.667855283636740, 10.19229649949594, 10.45829290067280, 11.15982605459676, 11.73627361489813, 12.25840456186237, 12.48224395814204, 13.12593603431703
