| L(s) = 1 | + 2-s + 4-s + 4·5-s − 7-s + 8-s + 4·10-s + 11-s + 2·13-s − 14-s + 16-s + 4·17-s − 6·19-s + 4·20-s + 22-s − 4·23-s + 11·25-s + 2·26-s − 28-s + 2·29-s − 2·31-s + 32-s + 4·34-s − 4·35-s + 10·37-s − 6·38-s + 4·40-s − 4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s + 0.353·8-s + 1.26·10-s + 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.894·20-s + 0.213·22-s − 0.834·23-s + 11/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s − 0.676·35-s + 1.64·37-s − 0.973·38-s + 0.632·40-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.553477576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.553477576\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.759674701374569314122222212467, −8.907037361397106316846384330744, −7.951642576699392563387512465158, −6.62445475159785453495873048483, −6.22501908736119603038057345164, −5.57547584921431243576094737475, −4.59118714632791292827191258849, −3.44639178218693714912433679777, −2.38911105843277843359160362418, −1.46845333940904453855522694316,
1.46845333940904453855522694316, 2.38911105843277843359160362418, 3.44639178218693714912433679777, 4.59118714632791292827191258849, 5.57547584921431243576094737475, 6.22501908736119603038057345164, 6.62445475159785453495873048483, 7.951642576699392563387512465158, 8.907037361397106316846384330744, 9.759674701374569314122222212467
