| L(s) = 1 | + 3·3-s + 5-s + 7-s + 6·9-s − 5·11-s + 3·13-s + 3·15-s − 17-s + 6·19-s + 3·21-s − 6·23-s + 25-s + 9·27-s + 9·29-s + 4·31-s − 15·33-s + 35-s − 2·37-s + 9·39-s − 4·41-s + 10·43-s + 6·45-s + 47-s + 49-s − 3·51-s − 4·53-s − 5·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 1.50·11-s + 0.832·13-s + 0.774·15-s − 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s + 0.718·31-s − 2.61·33-s + 0.169·35-s − 0.328·37-s + 1.44·39-s − 0.624·41-s + 1.52·43-s + 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.805093272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.805093272\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.861920653937128870724599342485, −8.213043350946324211142347413159, −7.85081335095312046952512435207, −6.96929537196328486214467340283, −5.86772395707333722143344910386, −4.92379386705551451390155657739, −3.98659668409059060417281652635, −2.96940713374575090223990451771, −2.44270886572835496551406086694, −1.32708580773053885811875668723,
1.32708580773053885811875668723, 2.44270886572835496551406086694, 2.96940713374575090223990451771, 3.98659668409059060417281652635, 4.92379386705551451390155657739, 5.86772395707333722143344910386, 6.96929537196328486214467340283, 7.85081335095312046952512435207, 8.213043350946324211142347413159, 8.861920653937128870724599342485
