| L(s) = 1 | − 2·2-s + 4-s − 2·7-s + 2·8-s − 2·9-s − 2·11-s + 4·14-s − 3·16-s + 4·18-s + 4·22-s − 8·23-s + 2·25-s − 2·28-s − 6·29-s − 2·32-s − 2·36-s − 10·37-s − 2·44-s + 16·46-s − 3·49-s − 4·50-s − 6·53-s − 4·56-s + 12·58-s + 4·63-s + 9·64-s + 6·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.755·7-s + 0.707·8-s − 2/3·9-s − 0.603·11-s + 1.06·14-s − 3/4·16-s + 0.942·18-s + 0.852·22-s − 1.66·23-s + 2/5·25-s − 0.377·28-s − 1.11·29-s − 0.353·32-s − 1/3·36-s − 1.64·37-s − 0.301·44-s + 2.35·46-s − 3/7·49-s − 0.565·50-s − 0.824·53-s − 0.534·56-s + 1.57·58-s + 0.503·63-s + 9/8·64-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−10.90540016677067739160058361458, −10.31992092186155275882524681776, −9.887274252426983216110948943680, −9.425962044952057250943894169053, −8.894696315209156475641507456161, −8.246096250127892399826289085723, −7.965590664277385752439484694374, −7.20821098072717542995738691737, −6.55922433122554832282790128488, −5.73489242527448112325495211812, −5.15813552350976083696828757737, −4.03416931780400837276619230590, −3.19693469563960983162401477685, −1.96155072902169868442403003992, 0,
1.96155072902169868442403003992, 3.19693469563960983162401477685, 4.03416931780400837276619230590, 5.15813552350976083696828757737, 5.73489242527448112325495211812, 6.55922433122554832282790128488, 7.20821098072717542995738691737, 7.965590664277385752439484694374, 8.246096250127892399826289085723, 8.894696315209156475641507456161, 9.425962044952057250943894169053, 9.887274252426983216110948943680, 10.31992092186155275882524681776, 10.90540016677067739160058361458
