| L(s) = 1 | − 2·2-s + 4·4-s + 20·7-s − 8·8-s − 27·9-s − 52·11-s + 42·13-s − 40·14-s + 16·16-s + 22·17-s + 54·18-s + 28·19-s + 104·22-s − 36·23-s − 84·26-s + 80·28-s + 29·29-s + 24·31-s − 32·32-s − 44·34-s − 108·36-s + 266·37-s − 56·38-s − 38·41-s − 88·43-s − 208·44-s + 72·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.07·7-s − 0.353·8-s − 9-s − 1.42·11-s + 0.896·13-s − 0.763·14-s + 1/4·16-s + 0.313·17-s + 0.707·18-s + 0.338·19-s + 1.00·22-s − 0.326·23-s − 0.633·26-s + 0.539·28-s + 0.185·29-s + 0.139·31-s − 0.176·32-s − 0.221·34-s − 1/2·36-s + 1.18·37-s − 0.239·38-s − 0.144·41-s − 0.312·43-s − 0.712·44-s + 0.230·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - p T \) |
| good | 3 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 36 T + p^{3} T^{2} \) |
| 31 | \( 1 - 24 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 38 T + p^{3} T^{2} \) |
| 43 | \( 1 + 88 T + p^{3} T^{2} \) |
| 47 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 53 | \( 1 - 194 T + p^{3} T^{2} \) |
| 59 | \( 1 + 460 T + p^{3} T^{2} \) |
| 61 | \( 1 + 314 T + p^{3} T^{2} \) |
| 67 | \( 1 + 896 T + p^{3} T^{2} \) |
| 71 | \( 1 + 416 T + p^{3} T^{2} \) |
| 73 | \( 1 - 606 T + p^{3} T^{2} \) |
| 79 | \( 1 - 992 T + p^{3} T^{2} \) |
| 83 | \( 1 - 24 T + p^{3} T^{2} \) |
| 89 | \( 1 + 774 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1626 T + p^{3} T^{2} \) |
| 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
−8.535645442926844696759955032806, −8.075262661165188828137793218307, −7.52479711900370116178459547522, −6.23023173779597733641697927173, −5.53314418307593661825019292918, −4.68647486118050043774788666035, −3.27547871726690779620442259434, −2.38768321369037320234003161467, −1.25004029532533717536016785532, 0,
1.25004029532533717536016785532, 2.38768321369037320234003161467, 3.27547871726690779620442259434, 4.68647486118050043774788666035, 5.53314418307593661825019292918, 6.23023173779597733641697927173, 7.52479711900370116178459547522, 8.075262661165188828137793218307, 8.535645442926844696759955032806
