| L(s) = 1 | + 2-s + 3·3-s + 3·6-s + 5·7-s − 8-s + 6·9-s − 3·11-s + 7·13-s + 5·14-s − 16-s − 3·17-s + 6·18-s − 7·19-s + 15·21-s − 3·22-s + 12·23-s − 3·24-s − 2·25-s + 7·26-s + 9·27-s − 3·29-s − 8·31-s − 9·33-s − 3·34-s − 12·37-s − 7·38-s + 21·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1.22·6-s + 1.88·7-s − 0.353·8-s + 2·9-s − 0.904·11-s + 1.94·13-s + 1.33·14-s − 1/4·16-s − 0.727·17-s + 1.41·18-s − 1.60·19-s + 3.27·21-s − 0.639·22-s + 2.50·23-s − 0.612·24-s − 2/5·25-s + 1.37·26-s + 1.73·27-s − 0.557·29-s − 1.43·31-s − 1.56·33-s − 0.514·34-s − 1.97·37-s − 1.13·38-s + 3.36·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.782905441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.782905441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84177431309224925357041016537, −10.82318833695876405541139087297, −10.30765734111193944929865928702, −9.507122661028961176946428581338, −8.954642786528935692383632500401, −8.583335330917070372761100024260, −8.399518132699041492276816647180, −8.358243979180356856401867568068, −7.29334407394395084745429439575, −7.20487540778183537262662955882, −6.66808747737944856848212619749, −5.67233096032637178991782036792, −5.35723241514589119346424560334, −4.85466274263137514455950931304, −4.07531622514595584239566222180, −3.98626857949496222011960525997, −3.28623842475651424666593322353, −2.55794422222255809473022854115, −1.94506129643303569782553429207, −1.40282037575030791985896656203,
1.40282037575030791985896656203, 1.94506129643303569782553429207, 2.55794422222255809473022854115, 3.28623842475651424666593322353, 3.98626857949496222011960525997, 4.07531622514595584239566222180, 4.85466274263137514455950931304, 5.35723241514589119346424560334, 5.67233096032637178991782036792, 6.66808747737944856848212619749, 7.20487540778183537262662955882, 7.29334407394395084745429439575, 8.358243979180356856401867568068, 8.399518132699041492276816647180, 8.583335330917070372761100024260, 8.954642786528935692383632500401, 9.507122661028961176946428581338, 10.30765734111193944929865928702, 10.82318833695876405541139087297, 10.84177431309224925357041016537
