| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s + 2·7-s − 4·8-s + 2·9-s − 4·10-s + 2·11-s + 6·12-s − 4·14-s + 4·15-s + 5·16-s + 2·17-s − 4·18-s + 6·20-s + 4·21-s − 4·22-s + 2·23-s − 8·24-s − 25-s + 6·27-s + 6·28-s − 10·29-s − 8·30-s − 6·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s + 0.603·11-s + 1.73·12-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.485·17-s − 0.942·18-s + 1.34·20-s + 0.872·21-s − 0.852·22-s + 0.417·23-s − 1.63·24-s − 1/5·25-s + 1.15·27-s + 1.13·28-s − 1.85·29-s − 1.46·30-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218593476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218593476\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−12.98297855813398328049697376026, −12.57770029277143203940798612033, −11.75835080359848482161474602238, −11.50550095622503468957749566825, −10.75585917222103581887500233158, −10.45411056985443318059315654876, −9.732840016200310498109780662634, −9.408660352072327745284470722577, −9.056645678738201117170623599350, −8.530638885889693371735128168558, −7.88266893666026732482711460922, −7.76002577991912302604844038393, −6.73006360911130935752534265473, −6.59311842322815813823235461349, −5.50429705291244738612543323185, −5.02158479735626705153511171456, −3.70619029077656432920417665381, −3.14425880851546873272857987591, −1.95183226077621285820039656705, −1.65870401694631459801251421648,
1.65870401694631459801251421648, 1.95183226077621285820039656705, 3.14425880851546873272857987591, 3.70619029077656432920417665381, 5.02158479735626705153511171456, 5.50429705291244738612543323185, 6.59311842322815813823235461349, 6.73006360911130935752534265473, 7.76002577991912302604844038393, 7.88266893666026732482711460922, 8.530638885889693371735128168558, 9.056645678738201117170623599350, 9.408660352072327745284470722577, 9.732840016200310498109780662634, 10.45411056985443318059315654876, 10.75585917222103581887500233158, 11.50550095622503468957749566825, 11.75835080359848482161474602238, 12.57770029277143203940798612033, 12.98297855813398328049697376026
