| L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s + 2·9-s − 8·11-s + 2·13-s + 4·15-s − 4·17-s + 2·19-s + 4·21-s − 4·23-s − 2·25-s − 6·27-s + 12·29-s + 16·33-s + 4·35-s + 4·37-s − 4·39-s − 4·41-s − 4·45-s + 8·47-s + 3·49-s + 8·51-s + 16·55-s − 4·57-s − 14·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s + 2/3·9-s − 2.41·11-s + 0.554·13-s + 1.03·15-s − 0.970·17-s + 0.458·19-s + 0.872·21-s − 0.834·23-s − 2/5·25-s − 1.15·27-s + 2.22·29-s + 2.78·33-s + 0.676·35-s + 0.657·37-s − 0.640·39-s − 0.624·41-s − 0.596·45-s + 1.16·47-s + 3/7·49-s + 1.12·51-s + 2.15·55-s − 0.529·57-s − 1.82·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.881014942611411619384651300057, −8.808368632162736066202741210694, −8.086577700238067299655157450306, −7.86160844803531547269126701210, −7.56633626263904897671942658996, −7.18990165055049598164849563608, −6.46441875304653880966488143750, −6.35250092801980494867377025217, −5.75461554987906457195854636981, −5.62483651115504444349883694516, −4.80711609506956019436257480220, −4.78506457139497289772517680456, −4.15730552800527270694336163481, −3.67256178251561749008942932086, −3.03429556676906260884683459961, −2.65590677722670881435561590895, −2.06135426230661848811073629963, −1.02333097575618114958165350669, 0, 0,
1.02333097575618114958165350669, 2.06135426230661848811073629963, 2.65590677722670881435561590895, 3.03429556676906260884683459961, 3.67256178251561749008942932086, 4.15730552800527270694336163481, 4.78506457139497289772517680456, 4.80711609506956019436257480220, 5.62483651115504444349883694516, 5.75461554987906457195854636981, 6.35250092801980494867377025217, 6.46441875304653880966488143750, 7.18990165055049598164849563608, 7.56633626263904897671942658996, 7.86160844803531547269126701210, 8.086577700238067299655157450306, 8.808368632162736066202741210694, 8.881014942611411619384651300057
