| L(s) = 1 | + 3-s + 5-s − 2·7-s − 2·9-s − 6·13-s + 15-s + 2·17-s + 6·19-s − 2·21-s − 6·25-s − 2·27-s − 5·31-s − 2·35-s − 2·37-s − 6·39-s − 3·41-s + 11·43-s − 2·45-s − 2·47-s + 3·49-s + 2·51-s + 11·53-s + 6·57-s + 4·59-s + 5·61-s + 4·63-s − 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s − 1.66·13-s + 0.258·15-s + 0.485·17-s + 1.37·19-s − 0.436·21-s − 6/5·25-s − 0.384·27-s − 0.898·31-s − 0.338·35-s − 0.328·37-s − 0.960·39-s − 0.468·41-s + 1.67·43-s − 0.298·45-s − 0.291·47-s + 3/7·49-s + 0.280·51-s + 1.51·53-s + 0.794·57-s + 0.520·59-s + 0.640·61-s + 0.503·63-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 65 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11 T + 113 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 133 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 143 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 26 T + 334 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 211 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.57536039375549332641684205213, −7.33174666925136630362952194753, −7.15032720808845486584219161969, −6.83922307129045932198163991185, −6.20408501780414235099559048568, −5.93297516164669373477741881550, −5.44162446925718712248612380019, −5.43868030621029244694009300299, −5.08475776723502891076083148394, −4.39461891552889317900838829156, −4.01242198527031390065436107263, −3.75194977994605370017860161274, −3.18259468819457691158692933965, −2.85562560309784182177288766388, −2.41870442446132389966969414472, −2.39167351789335839828403264526, −1.48001427681872157249579844859, −1.15898815137792494663967075663, 0, 0,
1.15898815137792494663967075663, 1.48001427681872157249579844859, 2.39167351789335839828403264526, 2.41870442446132389966969414472, 2.85562560309784182177288766388, 3.18259468819457691158692933965, 3.75194977994605370017860161274, 4.01242198527031390065436107263, 4.39461891552889317900838829156, 5.08475776723502891076083148394, 5.43868030621029244694009300299, 5.44162446925718712248612380019, 5.93297516164669373477741881550, 6.20408501780414235099559048568, 6.83922307129045932198163991185, 7.15032720808845486584219161969, 7.33174666925136630362952194753, 7.57536039375549332641684205213
