| L(s) = 1 | + 4·4-s + 5·9-s − 8·11-s + 12·16-s + 2·19-s + 20·29-s − 2·31-s + 20·36-s − 6·41-s − 32·44-s + 14·49-s − 22·59-s − 24·61-s + 32·64-s + 18·71-s + 8·76-s + 20·79-s + 16·81-s − 40·99-s − 14·101-s − 30·109-s + 80·116-s + 26·121-s − 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2·4-s + 5/3·9-s − 2.41·11-s + 3·16-s + 0.458·19-s + 3.71·29-s − 0.359·31-s + 10/3·36-s − 0.937·41-s − 4.82·44-s + 2·49-s − 2.86·59-s − 3.07·61-s + 4·64-s + 2.13·71-s + 0.917·76-s + 2.25·79-s + 16/9·81-s − 4.02·99-s − 1.39·101-s − 2.87·109-s + 7.42·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.742126061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.742126061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−10.48278345533446524992325430119, −10.41044303706651300394071596226, −9.904256373478041683973361706975, −9.466003611325508049518057608641, −8.722197186877822478329589311283, −7.890394711819109950625215206864, −7.88632026386161103566314793055, −7.70915822697217151511001502685, −6.84439527914552988698813881367, −6.83214982711235958026752157333, −6.28466943321559373227459233947, −5.76554771935643504569960634164, −5.00300429306524236642947842196, −4.94007675554633247580204355813, −4.12339276751089641210516377440, −3.25378305761747039797499486378, −2.81589308501884073096171315419, −2.48207714564487906477891693061, −1.70655469139794557059481133399, −1.03228321438965306959745188242,
1.03228321438965306959745188242, 1.70655469139794557059481133399, 2.48207714564487906477891693061, 2.81589308501884073096171315419, 3.25378305761747039797499486378, 4.12339276751089641210516377440, 4.94007675554633247580204355813, 5.00300429306524236642947842196, 5.76554771935643504569960634164, 6.28466943321559373227459233947, 6.83214982711235958026752157333, 6.84439527914552988698813881367, 7.70915822697217151511001502685, 7.88632026386161103566314793055, 7.890394711819109950625215206864, 8.722197186877822478329589311283, 9.466003611325508049518057608641, 9.904256373478041683973361706975, 10.41044303706651300394071596226, 10.48278345533446524992325430119
