| L(s) = 1 | − 3-s − 5-s − 2·7-s + 2·9-s − 2·13-s + 15-s − 9·17-s + 4·19-s + 2·21-s − 3·23-s − 25-s − 6·27-s − 3·31-s + 2·35-s − 2·37-s + 2·39-s + 10·41-s − 9·43-s − 2·45-s − 5·47-s + 2·49-s + 9·51-s + 5·53-s − 4·57-s − 2·59-s − 3·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 2/3·9-s − 0.554·13-s + 0.258·15-s − 2.18·17-s + 0.917·19-s + 0.436·21-s − 0.625·23-s − 1/5·25-s − 1.15·27-s − 0.538·31-s + 0.338·35-s − 0.328·37-s + 0.320·39-s + 1.56·41-s − 1.37·43-s − 0.298·45-s − 0.729·47-s + 2/7·49-s + 1.26·51-s + 0.686·53-s − 0.529·57-s − 0.260·59-s − 0.384·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| 103 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 80 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 82 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T - 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 150 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 19 T + 198 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
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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
−15.5413905684, −15.1671247140, −14.6940921871, −13.9648950448, −13.5088135648, −13.0686033832, −12.7919888999, −12.0590042299, −11.7709655845, −11.2591813167, −10.8068456909, −10.2215855981, −9.68678299080, −9.29395954134, −8.73635636591, −8.00530312244, −7.35343524271, −7.00966123895, −6.38433138858, −5.83873766706, −5.08703888738, −4.38045153417, −3.89204297680, −2.92819103786, −1.89907963087, 0,
1.89907963087, 2.92819103786, 3.89204297680, 4.38045153417, 5.08703888738, 5.83873766706, 6.38433138858, 7.00966123895, 7.35343524271, 8.00530312244, 8.73635636591, 9.29395954134, 9.68678299080, 10.2215855981, 10.8068456909, 11.2591813167, 11.7709655845, 12.0590042299, 12.7919888999, 13.0686033832, 13.5088135648, 13.9648950448, 14.6940921871, 15.1671247140, 15.5413905684
