| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4·7-s − 8-s + 9-s − 10-s − 12-s + 2·13-s + 4·14-s − 15-s + 16-s − 18-s − 4·19-s + 20-s + 4·21-s + 24-s + 25-s − 2·26-s − 27-s − 4·28-s − 10·29-s + 30-s + 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.182·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.141035665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.141035665\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.42251754876443, −12.00198588190687, −11.62325674367987, −10.90742605376485, −10.63781204893989, −10.26832685333028, −9.748247172568561, −9.354229656483381, −9.060885700335854, −8.446000445380870, −7.981411918527076, −7.372654768086343, −6.746092169862888, −6.608407546688627, −6.108362556772598, −5.658156145494130, −5.254138250954707, −4.361348698525465, −3.976454127141903, −3.262087829830439, −2.950439690387401, −2.056671443677738, −1.803955690952759, −0.7863373910517596, −0.4295994226743138,
0.4295994226743138, 0.7863373910517596, 1.803955690952759, 2.056671443677738, 2.950439690387401, 3.262087829830439, 3.976454127141903, 4.361348698525465, 5.254138250954707, 5.658156145494130, 6.108362556772598, 6.608407546688627, 6.746092169862888, 7.372654768086343, 7.981411918527076, 8.446000445380870, 9.060885700335854, 9.354229656483381, 9.748247172568561, 10.26832685333028, 10.63781204893989, 10.90742605376485, 11.62325674367987, 12.00198588190687, 12.42251754876443
