| L(s) = 1 | + 2-s + 3·3-s − 5-s + 3·6-s + 5·7-s − 8-s + 3·9-s − 10-s + 11-s − 12·13-s + 5·14-s − 3·15-s − 16-s + 8·17-s + 3·18-s + 4·19-s + 15·21-s + 22-s + 7·23-s − 3·24-s − 12·26-s − 18·29-s − 3·30-s + 6·31-s + 3·33-s + 8·34-s − 5·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 0.447·5-s + 1.22·6-s + 1.88·7-s − 0.353·8-s + 9-s − 0.316·10-s + 0.301·11-s − 3.32·13-s + 1.33·14-s − 0.774·15-s − 1/4·16-s + 1.94·17-s + 0.707·18-s + 0.917·19-s + 3.27·21-s + 0.213·22-s + 1.45·23-s − 0.612·24-s − 2.35·26-s − 3.34·29-s − 0.547·30-s + 1.07·31-s + 0.522·33-s + 1.37·34-s − 0.845·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.334079296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.334079296\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−10.47472943755877980269536698809, −9.803023084406188171713769170375, −9.707035204662238984669709962604, −9.081103876239041146506428432676, −8.990575632617774788828346030889, −8.225780975980148185478388689080, −7.86792319509217213922730209397, −7.58452757228163365624984660536, −7.24425746640694213488687131029, −7.15855935577966827901739890523, −5.70938776108141668318386144848, −5.53120101770061979353336296781, −4.91656050240936604502235196627, −4.79132410350599907082174335612, −4.06338766758360249651744973537, −3.50929864174812128772114031512, −3.03421876184137734924393530905, −2.37315634130758888854816437791, −2.13519828493369646656771448880, −0.999832691472777230762962001553,
0.999832691472777230762962001553, 2.13519828493369646656771448880, 2.37315634130758888854816437791, 3.03421876184137734924393530905, 3.50929864174812128772114031512, 4.06338766758360249651744973537, 4.79132410350599907082174335612, 4.91656050240936604502235196627, 5.53120101770061979353336296781, 5.70938776108141668318386144848, 7.15855935577966827901739890523, 7.24425746640694213488687131029, 7.58452757228163365624984660536, 7.86792319509217213922730209397, 8.225780975980148185478388689080, 8.990575632617774788828346030889, 9.081103876239041146506428432676, 9.707035204662238984669709962604, 9.803023084406188171713769170375, 10.47472943755877980269536698809
