| L(s) = 1 | − 2·3-s − 4-s − 5-s − 3·7-s + 2·9-s + 5·11-s + 2·12-s − 13-s + 2·15-s + 16-s + 19-s + 20-s + 6·21-s + 9·23-s − 2·25-s − 6·27-s + 3·28-s − 6·29-s − 6·31-s − 10·33-s + 3·35-s − 2·36-s − 37-s + 2·39-s + 7·41-s − 5·44-s − 2·45-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.447·5-s − 1.13·7-s + 2/3·9-s + 1.50·11-s + 0.577·12-s − 0.277·13-s + 0.516·15-s + 1/4·16-s + 0.229·19-s + 0.223·20-s + 1.30·21-s + 1.87·23-s − 2/5·25-s − 1.15·27-s + 0.566·28-s − 1.11·29-s − 1.07·31-s − 1.74·33-s + 0.507·35-s − 1/3·36-s − 0.164·37-s + 0.320·39-s + 1.09·41-s − 0.753·44-s − 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) |
| 193 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 42 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 38 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 76 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 156 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−15.0664538120, −14.7091307611, −14.5292710340, −13.5962764031, −13.2084793599, −12.8838677148, −12.4409264396, −11.8480176773, −11.4993099618, −11.1652021085, −10.5869594758, −9.93142648551, −9.42233391663, −9.15362086146, −8.70285914751, −7.59040138312, −7.34009599924, −6.70330340609, −6.15651839500, −5.69380980306, −5.03329285582, −4.30013680022, −3.70895329641, −3.08581889157, −1.45755306521, 0,
1.45755306521, 3.08581889157, 3.70895329641, 4.30013680022, 5.03329285582, 5.69380980306, 6.15651839500, 6.70330340609, 7.34009599924, 7.59040138312, 8.70285914751, 9.15362086146, 9.42233391663, 9.93142648551, 10.5869594758, 11.1652021085, 11.4993099618, 11.8480176773, 12.4409264396, 12.8838677148, 13.2084793599, 13.5962764031, 14.5292710340, 14.7091307611, 15.0664538120
