| L(s) = 1 | − 5-s + 7-s + 9-s + 17-s − 23-s + 25-s − 29-s − 31-s − 35-s + 37-s − 41-s − 2·43-s − 45-s + 53-s − 59-s + 63-s + 67-s − 71-s + 81-s + 83-s − 85-s − 2·97-s − 101-s − 2·103-s + 107-s + 113-s + 115-s + ⋯ |
| L(s) = 1 | − 5-s + 7-s + 9-s + 17-s − 23-s + 25-s − 29-s − 31-s − 35-s + 37-s − 41-s − 2·43-s − 45-s + 53-s − 59-s + 63-s + 67-s − 71-s + 81-s + 83-s − 85-s − 2·97-s − 101-s − 2·103-s + 107-s + 113-s + 115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8577521555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8577521555\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−11.36952701813825030829229209052, −10.46711075154924793766312378312, −9.552403465041702388085711402385, −8.283411583675314545199774185430, −7.74386916968499461914545689246, −6.89910560101402597985849910649, −5.42909400221107815627451061321, −4.42997744248027076135827739520, −3.52927042259847306133125452764, −1.64965578133303113593360009657,
1.64965578133303113593360009657, 3.52927042259847306133125452764, 4.42997744248027076135827739520, 5.42909400221107815627451061321, 6.89910560101402597985849910649, 7.74386916968499461914545689246, 8.283411583675314545199774185430, 9.552403465041702388085711402385, 10.46711075154924793766312378312, 11.36952701813825030829229209052
