| L(s) = 1 | − 2·2-s + 8·8-s + 36·13-s − 16·16-s − 20·17-s − 72·26-s + 72·29-s + 40·34-s + 108·37-s − 36·41-s − 10·49-s + 52·53-s − 144·58-s − 148·61-s + 64·64-s + 72·73-s − 216·74-s + 72·82-s + 36·89-s − 144·97-s + 20·98-s − 72·101-s + 288·104-s − 104·106-s − 52·109-s + 20·113-s + 134·121-s + ⋯ |
| L(s) = 1 | − 2-s + 8-s + 2.76·13-s − 16-s − 1.17·17-s − 2.76·26-s + 2.48·29-s + 1.17·34-s + 2.91·37-s − 0.878·41-s − 0.204·49-s + 0.981·53-s − 2.48·58-s − 2.42·61-s + 64-s + 0.986·73-s − 2.91·74-s + 0.878·82-s + 0.404·89-s − 1.48·97-s + 0.204·98-s − 0.712·101-s + 2.76·104-s − 0.981·106-s − 0.477·109-s + 0.176·113-s + 1.10·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.707983285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707983285\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 10 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 134 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1010 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5990 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7250 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 718 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 4370 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5666 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 72 T + p^{2} T^{2} )^{2} \) |
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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
−9.981751637693959073978559135186, −9.773719138048482954677431317244, −8.990590261134404974295501635210, −8.953721199796506089605513759842, −8.551912277390231074184078548440, −8.042357073702417448344540208087, −7.954042425382920447742641694216, −7.25043517686465679580226277914, −6.48519449670707884957183400033, −6.45394860019986659299713521235, −6.06359477420509383610556088561, −5.30212225250317417093816116493, −4.71905351340388600562903938607, −4.13001374729030913117272241390, −4.05260414601340605292945754260, −3.11534711763170037412413876757, −2.61787344751079124540848566845, −1.69064483572484304016050497037, −1.13885131245011505877905370491, −0.61535999255230883171405874797,
0.61535999255230883171405874797, 1.13885131245011505877905370491, 1.69064483572484304016050497037, 2.61787344751079124540848566845, 3.11534711763170037412413876757, 4.05260414601340605292945754260, 4.13001374729030913117272241390, 4.71905351340388600562903938607, 5.30212225250317417093816116493, 6.06359477420509383610556088561, 6.45394860019986659299713521235, 6.48519449670707884957183400033, 7.25043517686465679580226277914, 7.954042425382920447742641694216, 8.042357073702417448344540208087, 8.551912277390231074184078548440, 8.953721199796506089605513759842, 8.990590261134404974295501635210, 9.773719138048482954677431317244, 9.981751637693959073978559135186
