| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s + 4·8-s + 3·9-s − 4·10-s − 10·11-s + 6·12-s − 2·13-s − 4·15-s + 5·16-s − 2·17-s + 6·18-s + 2·19-s − 6·20-s − 20·22-s − 10·23-s + 8·24-s − 7·25-s − 4·26-s + 4·27-s − 2·29-s − 8·30-s + 6·32-s − 20·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s − 3.01·11-s + 1.73·12-s − 0.554·13-s − 1.03·15-s + 5/4·16-s − 0.485·17-s + 1.41·18-s + 0.458·19-s − 1.34·20-s − 4.26·22-s − 2.08·23-s + 1.63·24-s − 7/5·25-s − 0.784·26-s + 0.769·27-s − 0.371·29-s − 1.46·30-s + 1.06·32-s − 3.48·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 121 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 201 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 108 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 224 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.147236032374330448322800267331, −7.952021380257314892208085088438, −7.46660151077778050334636802937, −7.43556954227498826490847724191, −6.78198743398191569725072535838, −6.69321178483256316094987064104, −5.73295017213657611597707207199, −5.61106965398206116498493454281, −5.19351025338025645557675936994, −5.00649849469277909624954076827, −4.32465467240853572498843441969, −4.05759312995721582579470629434, −3.62647412958882658047175700611, −3.34183576380681669974137440925, −2.71958445992661757566858906348, −2.54760596510965282736806492296, −1.86682565985406163731925913206, −1.80760691237883472289884600279, 0, 0,
1.80760691237883472289884600279, 1.86682565985406163731925913206, 2.54760596510965282736806492296, 2.71958445992661757566858906348, 3.34183576380681669974137440925, 3.62647412958882658047175700611, 4.05759312995721582579470629434, 4.32465467240853572498843441969, 5.00649849469277909624954076827, 5.19351025338025645557675936994, 5.61106965398206116498493454281, 5.73295017213657611597707207199, 6.69321178483256316094987064104, 6.78198743398191569725072535838, 7.43556954227498826490847724191, 7.46660151077778050334636802937, 7.952021380257314892208085088438, 8.147236032374330448322800267331
