| L(s) = 1 | − 3·3-s − 5-s + 3·9-s + 2·11-s + 12·13-s + 3·15-s + 2·17-s + 9·23-s + 6·29-s + 2·31-s − 6·33-s − 8·37-s − 36·39-s − 10·41-s + 2·43-s − 3·45-s + 8·47-s − 6·51-s − 4·53-s − 2·55-s − 8·59-s + 7·61-s − 12·65-s + 3·67-s − 27·69-s + 16·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 9-s + 0.603·11-s + 3.32·13-s + 0.774·15-s + 0.485·17-s + 1.87·23-s + 1.11·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s − 5.76·39-s − 1.56·41-s + 0.304·43-s − 0.447·45-s + 1.16·47-s − 0.840·51-s − 0.549·53-s − 0.269·55-s − 1.04·59-s + 0.896·61-s − 1.48·65-s + 0.366·67-s − 3.25·69-s + 1.89·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402513147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402513147\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.31129931620928935996612885302, −10.12415045780528937101596296056, −9.140461885245802917817875130940, −9.019429307288767411934948838560, −8.575985217166682750136144618897, −8.199973389632290018009885314443, −7.74886919339459238350470747174, −6.89064595664227616231617602403, −6.69752257954642342614077714698, −6.40432213967838869900548633656, −5.83866256231900707754218541761, −5.66851783944335284476522793276, −4.95869957540646618009965506755, −4.74011430270501850512274903563, −3.90327264761187513524774144251, −3.35323353790531445569869501298, −3.35284352103996439862655159452, −1.96222052862298600103783583546, −0.998071215017101942980472863605, −0.869566947810801854820645664028,
0.869566947810801854820645664028, 0.998071215017101942980472863605, 1.96222052862298600103783583546, 3.35284352103996439862655159452, 3.35323353790531445569869501298, 3.90327264761187513524774144251, 4.74011430270501850512274903563, 4.95869957540646618009965506755, 5.66851783944335284476522793276, 5.83866256231900707754218541761, 6.40432213967838869900548633656, 6.69752257954642342614077714698, 6.89064595664227616231617602403, 7.74886919339459238350470747174, 8.199973389632290018009885314443, 8.575985217166682750136144618897, 9.019429307288767411934948838560, 9.140461885245802917817875130940, 10.12415045780528937101596296056, 10.31129931620928935996612885302
