| L(s) = 1 | + 4-s − 5-s + 2·7-s + 2·13-s − 3·16-s − 3·17-s − 19-s − 20-s − 11·23-s − 8·25-s + 2·28-s + 12·29-s + 5·31-s − 2·35-s + 7·37-s − 17·41-s + 6·43-s + 3·49-s + 2·52-s − 18·53-s − 2·59-s − 7·64-s − 2·65-s − 3·68-s − 8·71-s + 14·73-s − 76-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.554·13-s − 3/4·16-s − 0.727·17-s − 0.229·19-s − 0.223·20-s − 2.29·23-s − 8/5·25-s + 0.377·28-s + 2.22·29-s + 0.898·31-s − 0.338·35-s + 1.15·37-s − 2.65·41-s + 0.914·43-s + 3/7·49-s + 0.277·52-s − 2.47·53-s − 0.260·59-s − 7/8·64-s − 0.248·65-s − 0.363·68-s − 0.949·71-s + 1.63·73-s − 0.114·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 11 T + 75 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 85 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 17 T + 153 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 197 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.82504366911934375959338371316, −7.35397365692341344944584635532, −6.99637478132851251699325244865, −6.58703213407774613901806172964, −6.38199236330212949739430568276, −6.04239184318070980121968899899, −5.70941443819596765643991295399, −5.28582640359934451469141683785, −4.66343163439297329573540439333, −4.40790285380491469658246984458, −4.26126498243089773543364637937, −3.89891644587520754339340644751, −3.28855809197973904528713209183, −2.72230315961593510172873749443, −2.60712715457247415344601303352, −1.90173726130557904002891758693, −1.62929801711823748842356098569, −1.18866396551325227321858583292, 0, 0,
1.18866396551325227321858583292, 1.62929801711823748842356098569, 1.90173726130557904002891758693, 2.60712715457247415344601303352, 2.72230315961593510172873749443, 3.28855809197973904528713209183, 3.89891644587520754339340644751, 4.26126498243089773543364637937, 4.40790285380491469658246984458, 4.66343163439297329573540439333, 5.28582640359934451469141683785, 5.70941443819596765643991295399, 6.04239184318070980121968899899, 6.38199236330212949739430568276, 6.58703213407774613901806172964, 6.99637478132851251699325244865, 7.35397365692341344944584635532, 7.82504366911934375959338371316
