| L(s) = 1 | − 2·2-s − 3-s + 3·4-s − 5-s + 2·6-s − 4·7-s − 4·8-s + 3·9-s + 2·10-s + 8·11-s − 3·12-s + 2·13-s + 8·14-s + 15-s + 5·16-s + 4·17-s − 6·18-s − 6·19-s − 3·20-s + 4·21-s − 16·22-s + 4·24-s − 4·26-s − 8·27-s − 12·28-s + 29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s + 0.816·6-s − 1.51·7-s − 1.41·8-s + 9-s + 0.632·10-s + 2.41·11-s − 0.866·12-s + 0.554·13-s + 2.13·14-s + 0.258·15-s + 5/4·16-s + 0.970·17-s − 1.41·18-s − 1.37·19-s − 0.670·20-s + 0.872·21-s − 3.41·22-s + 0.816·24-s − 0.784·26-s − 1.53·27-s − 2.26·28-s + 0.185·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6014682591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6014682591\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 43 | $C_2$ | \( 1 - 13 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−11.19042147010364679801769026593, −10.90357690617237070257372050998, −10.49256986122897986728305593332, −9.801553008570871737418492664523, −9.491090881638428952065518260695, −9.387707645537724001207292507419, −8.769571002863177817159551211057, −8.374805229028079741615386775442, −7.60632352904716918415103119352, −7.15547121294777060635087937328, −6.85513373271234795249067329002, −6.38042348501784897460179729649, −5.97546668793597634484490527596, −5.54290316381528571667639790313, −4.20709699562447033373098647536, −3.82864590605153507172563105819, −3.59323796336000712619914678141, −2.39424060279181679936751943070, −1.49611383046365336995502826650, −0.69760740524211949327848596101,
0.69760740524211949327848596101, 1.49611383046365336995502826650, 2.39424060279181679936751943070, 3.59323796336000712619914678141, 3.82864590605153507172563105819, 4.20709699562447033373098647536, 5.54290316381528571667639790313, 5.97546668793597634484490527596, 6.38042348501784897460179729649, 6.85513373271234795249067329002, 7.15547121294777060635087937328, 7.60632352904716918415103119352, 8.374805229028079741615386775442, 8.769571002863177817159551211057, 9.387707645537724001207292507419, 9.491090881638428952065518260695, 9.801553008570871737418492664523, 10.49256986122897986728305593332, 10.90357690617237070257372050998, 11.19042147010364679801769026593
