| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 5·7-s + 4·8-s − 2·9-s − 5·11-s − 3·12-s + 10·14-s + 5·16-s − 5·17-s − 4·18-s − 8·19-s − 5·21-s − 10·22-s − 4·23-s − 4·24-s + 2·27-s + 15·28-s + 3·29-s − 10·31-s + 6·32-s + 5·33-s − 10·34-s − 6·36-s − 11·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 1.88·7-s + 1.41·8-s − 2/3·9-s − 1.50·11-s − 0.866·12-s + 2.67·14-s + 5/4·16-s − 1.21·17-s − 0.942·18-s − 1.83·19-s − 1.09·21-s − 2.13·22-s − 0.834·23-s − 0.816·24-s + 0.384·27-s + 2.83·28-s + 0.557·29-s − 1.79·31-s + 1.06·32-s + 0.870·33-s − 1.71·34-s − 36-s − 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 5 T + 11 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 57 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 101 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 17 T + 163 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 75 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 125 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 211 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 11 T + 179 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 27 T + 373 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
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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
−7.51450049278844857236668677597, −7.37544245568357376033155831781, −6.80681297479685182981834850005, −6.41602582247317863418965067589, −5.98978107902284395080357976957, −5.93085451771711474945478923830, −5.38794059660175776440416438628, −5.16127180428494891982886045952, −4.81278253404303884356745896962, −4.63480596595810505060462216207, −4.07953380790296540504814325336, −4.03531204025034653569791743130, −3.25144941070123142293037839260, −2.97037924454120474783034153864, −2.19182608797616264145907164010, −2.18208922936816804170571726153, −1.88055600778629157656111326281, −1.21646978119891579217813172351, 0, 0,
1.21646978119891579217813172351, 1.88055600778629157656111326281, 2.18208922936816804170571726153, 2.19182608797616264145907164010, 2.97037924454120474783034153864, 3.25144941070123142293037839260, 4.03531204025034653569791743130, 4.07953380790296540504814325336, 4.63480596595810505060462216207, 4.81278253404303884356745896962, 5.16127180428494891982886045952, 5.38794059660175776440416438628, 5.93085451771711474945478923830, 5.98978107902284395080357976957, 6.41602582247317863418965067589, 6.80681297479685182981834850005, 7.37544245568357376033155831781, 7.51450049278844857236668677597
