| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s + 6·7-s + 4·8-s + 3·9-s − 4·10-s − 4·11-s − 6·12-s + 12·14-s + 4·15-s + 5·16-s − 8·17-s + 6·18-s − 8·19-s − 6·20-s − 12·21-s − 8·22-s − 8·24-s + 3·25-s − 4·27-s + 18·28-s − 4·29-s + 8·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 2.26·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.20·11-s − 1.73·12-s + 3.20·14-s + 1.03·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s − 1.83·19-s − 1.34·20-s − 2.61·21-s − 1.70·22-s − 1.63·24-s + 3/5·25-s − 0.769·27-s + 3.40·28-s − 0.742·29-s + 1.46·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 107 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 170 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.935552681287515422936024267515, −7.88299866769248219352661022134, −7.01848734756757079431531489282, −6.94710603968099400021566230738, −6.42536527231317676431691783827, −6.40580164335150059822681031794, −5.60360310655900281035659718470, −5.43769109386473999860857391082, −4.87552526433577389126785667934, −4.71569749720832739406751425049, −4.46185716696408108821753984839, −4.44179560312613737490121795082, −3.54140772495411128304689791254, −3.41696956438756135130691268294, −2.41404428317408817802251251877, −2.30628217005516636443956823120, −1.56333367285214444529380078693, −1.50772829219478187368550182815, 0, 0,
1.50772829219478187368550182815, 1.56333367285214444529380078693, 2.30628217005516636443956823120, 2.41404428317408817802251251877, 3.41696956438756135130691268294, 3.54140772495411128304689791254, 4.44179560312613737490121795082, 4.46185716696408108821753984839, 4.71569749720832739406751425049, 4.87552526433577389126785667934, 5.43769109386473999860857391082, 5.60360310655900281035659718470, 6.40580164335150059822681031794, 6.42536527231317676431691783827, 6.94710603968099400021566230738, 7.01848734756757079431531489282, 7.88299866769248219352661022134, 7.935552681287515422936024267515
