| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s + 2·7-s − 4·8-s − 4·10-s + 2·11-s + 2·13-s − 4·14-s + 8·16-s + 6·17-s − 10·19-s + 4·20-s − 4·22-s − 10·23-s + 3·25-s − 4·26-s + 4·28-s − 8·29-s − 14·31-s − 8·32-s − 12·34-s + 4·35-s + 2·37-s + 20·38-s − 8·40-s − 2·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s + 0.755·7-s − 1.41·8-s − 1.26·10-s + 0.603·11-s + 0.554·13-s − 1.06·14-s + 2·16-s + 1.45·17-s − 2.29·19-s + 0.894·20-s − 0.852·22-s − 2.08·23-s + 3/5·25-s − 0.784·26-s + 0.755·28-s − 1.48·29-s − 2.51·31-s − 1.41·32-s − 2.05·34-s + 0.676·35-s + 0.328·37-s + 3.24·38-s − 1.26·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12006225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12006225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 51 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 71 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 95 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 24 T + 259 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 155 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 139 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 207 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.450769243367842638330331465685, −8.065407776510532093839153858023, −7.78382536356182733500331059295, −7.66189841584485728434900570870, −6.83288090840068274112315226912, −6.57357558397370905276555626252, −6.27874219302585509265440313839, −5.69578107279564780223258428544, −5.55682464769454182896073152541, −5.39869346218672891639910679746, −4.29907252782526253377472981168, −4.16716929164566352906039846776, −3.70642906090214743588579595325, −3.10683311182927144684045958711, −2.62687083849799696594362132525, −1.89363901214474185200929657476, −1.54297570215907051646939419224, −1.47512616450842184737927867608, 0, 0,
1.47512616450842184737927867608, 1.54297570215907051646939419224, 1.89363901214474185200929657476, 2.62687083849799696594362132525, 3.10683311182927144684045958711, 3.70642906090214743588579595325, 4.16716929164566352906039846776, 4.29907252782526253377472981168, 5.39869346218672891639910679746, 5.55682464769454182896073152541, 5.69578107279564780223258428544, 6.27874219302585509265440313839, 6.57357558397370905276555626252, 6.83288090840068274112315226912, 7.66189841584485728434900570870, 7.78382536356182733500331059295, 8.065407776510532093839153858023, 8.450769243367842638330331465685
