| L(s) = 1 | + 2-s + 2·3-s − 2·4-s + 2·6-s − 3·8-s + 2·9-s + 4·11-s − 4·12-s + 2·13-s + 16-s + 4·17-s + 2·18-s + 4·22-s + 8·23-s − 6·24-s + 2·26-s + 6·27-s + 10·29-s + 6·31-s + 2·32-s + 8·33-s + 4·34-s − 4·36-s + 6·37-s + 4·39-s − 14·41-s + 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s − 1.06·8-s + 2/3·9-s + 1.20·11-s − 1.15·12-s + 0.554·13-s + 1/4·16-s + 0.970·17-s + 0.471·18-s + 0.852·22-s + 1.66·23-s − 1.22·24-s + 0.392·26-s + 1.15·27-s + 1.85·29-s + 1.07·31-s + 0.353·32-s + 1.39·33-s + 0.685·34-s − 2/3·36-s + 0.986·37-s + 0.640·39-s − 2.18·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.893056180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.893056180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 133 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 262 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 30 T + 398 T^{2} + 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 198 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.818904250915282455973289348366, −9.493363259432632079369868474619, −8.890918003524942691538104282779, −8.827416866654590100047171522799, −8.327507700963415256516021348043, −8.200045315219674707142203787692, −7.53881528940075562291240651760, −6.91584616837430072010938452916, −6.63979598166984275328347235209, −6.25854324354267974774304285954, −5.49780596409646562229034315239, −5.19643903778862967809830211199, −4.60051036166886667380316039219, −4.34548315632484463919046543122, −3.76510295175990068113821448737, −3.43933625865169725969500505618, −2.85540013776286813606637232234, −2.52401877585253848364795705257, −1.21924125463355925731094325499, −1.01615396283008186884136936959,
1.01615396283008186884136936959, 1.21924125463355925731094325499, 2.52401877585253848364795705257, 2.85540013776286813606637232234, 3.43933625865169725969500505618, 3.76510295175990068113821448737, 4.34548315632484463919046543122, 4.60051036166886667380316039219, 5.19643903778862967809830211199, 5.49780596409646562229034315239, 6.25854324354267974774304285954, 6.63979598166984275328347235209, 6.91584616837430072010938452916, 7.53881528940075562291240651760, 8.200045315219674707142203787692, 8.327507700963415256516021348043, 8.827416866654590100047171522799, 8.890918003524942691538104282779, 9.493363259432632079369868474619, 9.818904250915282455973289348366
