| L(s) = 1 | − 2·2-s + 4-s + 3·7-s − 13-s − 6·14-s + 16-s − 8·17-s − 7·19-s + 25-s + 2·26-s + 3·28-s + 6·29-s + 7·31-s + 2·32-s + 16·34-s + 6·37-s + 14·38-s − 3·41-s − 43-s − 47-s + 2·49-s − 2·50-s − 52-s − 2·53-s − 12·58-s − 5·59-s − 7·61-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.13·7-s − 0.277·13-s − 1.60·14-s + 1/4·16-s − 1.94·17-s − 1.60·19-s + 1/5·25-s + 0.392·26-s + 0.566·28-s + 1.11·29-s + 1.25·31-s + 0.353·32-s + 2.74·34-s + 0.986·37-s + 2.27·38-s − 0.468·41-s − 0.152·43-s − 0.145·47-s + 2/7·49-s − 0.282·50-s − 0.138·52-s − 0.274·53-s − 1.57·58-s − 0.650·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 4013 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 83 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 103 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 21 T + 245 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 3 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 172 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−14.4843222278, −13.6696957700, −13.5079517610, −13.0217340345, −12.3477030036, −11.9953431733, −11.4873044049, −10.9621390683, −10.7073695610, −10.2302851970, −9.70908429570, −9.20257312162, −8.74568782093, −8.41036856876, −8.16532575253, −7.60459843109, −6.84964257506, −6.42955876414, −5.98322017496, −4.90248130383, −4.55576588026, −4.20290195031, −2.87411233258, −2.26428067794, −1.33645078900, 0,
1.33645078900, 2.26428067794, 2.87411233258, 4.20290195031, 4.55576588026, 4.90248130383, 5.98322017496, 6.42955876414, 6.84964257506, 7.60459843109, 8.16532575253, 8.41036856876, 8.74568782093, 9.20257312162, 9.70908429570, 10.2302851970, 10.7073695610, 10.9621390683, 11.4873044049, 11.9953431733, 12.3477030036, 13.0217340345, 13.5079517610, 13.6696957700, 14.4843222278
