| L(s) = 1 | − 2-s − 3-s + 6-s − 5·7-s + 8-s − 3·11-s + 5·13-s + 5·14-s − 16-s + 4·17-s + 19-s + 5·21-s + 3·22-s − 24-s − 5·26-s + 27-s − 2·29-s + 8·31-s + 3·33-s − 4·34-s + 9·37-s − 38-s − 5·39-s − 10·41-s − 5·42-s − 12·43-s − 14·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 1.88·7-s + 0.353·8-s − 0.904·11-s + 1.38·13-s + 1.33·14-s − 1/4·16-s + 0.970·17-s + 0.229·19-s + 1.09·21-s + 0.639·22-s − 0.204·24-s − 0.980·26-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.522·33-s − 0.685·34-s + 1.47·37-s − 0.162·38-s − 0.800·39-s − 1.56·41-s − 0.771·42-s − 1.82·43-s − 2.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.860755367374226912169128731078, −8.633668730682568768718272982147, −8.342787922219346291239794253645, −7.919038864513036765720519041166, −7.33433385074875466327808609852, −7.12682480182311245929026661829, −6.50614865923428790735502677414, −6.24281428193857265684254673940, −5.83988468061097728062393147108, −5.70096774083562931125880213594, −4.79640447058617618700951501828, −4.68789934629655879285196644494, −3.95904194807670401985306052421, −3.20618591904642399834836365083, −3.17323460067459020427447575261, −2.77924599510267735893314031634, −1.53415693430852163512524651343, −1.29507049863174431134276148192, 0, 0,
1.29507049863174431134276148192, 1.53415693430852163512524651343, 2.77924599510267735893314031634, 3.17323460067459020427447575261, 3.20618591904642399834836365083, 3.95904194807670401985306052421, 4.68789934629655879285196644494, 4.79640447058617618700951501828, 5.70096774083562931125880213594, 5.83988468061097728062393147108, 6.24281428193857265684254673940, 6.50614865923428790735502677414, 7.12682480182311245929026661829, 7.33433385074875466327808609852, 7.919038864513036765720519041166, 8.342787922219346291239794253645, 8.633668730682568768718272982147, 8.860755367374226912169128731078
