| L(s) = 1 | + 3·3-s − 2·5-s + 5·7-s + 6·9-s + 4·11-s − 3·13-s − 6·15-s − 7·17-s + 5·19-s + 15·21-s + 4·23-s + 5·25-s + 9·27-s + 29-s + 6·31-s + 12·33-s − 10·35-s − 11·37-s − 9·39-s + 9·41-s + 5·43-s − 12·45-s − 6·47-s + 18·49-s − 21·51-s − 3·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s + 1.88·7-s + 2·9-s + 1.20·11-s − 0.832·13-s − 1.54·15-s − 1.69·17-s + 1.14·19-s + 3.27·21-s + 0.834·23-s + 25-s + 1.73·27-s + 0.185·29-s + 1.07·31-s + 2.08·33-s − 1.69·35-s − 1.80·37-s − 1.44·39-s + 1.40·41-s + 0.762·43-s − 1.78·45-s − 0.875·47-s + 18/7·49-s − 2.94·51-s − 0.412·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.012515122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.012515122\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−10.04987011868567999965261103077, −9.580623764775333824206864808598, −9.010229581998222518991430145134, −8.983098833843363175236530923531, −8.465082559638128021093899820364, −8.264224800792894946871294977982, −7.56357201799721972521484114285, −7.55064606853213380322955324923, −6.92362713723794364108264322778, −6.77816189079914069289621038942, −5.91065836003762704992609583981, −5.02407554929937565545323275160, −4.73635142531310905906069392147, −4.52812897853329639700568417339, −3.69392449041097977384461833889, −3.68779300759520933664229110864, −2.53699640437247579498669236356, −2.51481583543635874043120529099, −1.57801668411419504279734297154, −1.05508545308900066179799954172,
1.05508545308900066179799954172, 1.57801668411419504279734297154, 2.51481583543635874043120529099, 2.53699640437247579498669236356, 3.68779300759520933664229110864, 3.69392449041097977384461833889, 4.52812897853329639700568417339, 4.73635142531310905906069392147, 5.02407554929937565545323275160, 5.91065836003762704992609583981, 6.77816189079914069289621038942, 6.92362713723794364108264322778, 7.55064606853213380322955324923, 7.56357201799721972521484114285, 8.264224800792894946871294977982, 8.465082559638128021093899820364, 8.983098833843363175236530923531, 9.010229581998222518991430145134, 9.580623764775333824206864808598, 10.04987011868567999965261103077
