| L(s) = 1 | + 2-s + 7-s − 8-s + 11-s + 14-s − 16-s + 22-s + 23-s + 29-s − 37-s − 43-s + 46-s + 49-s − 2·53-s − 56-s + 58-s + 64-s − 67-s + 71-s − 74-s + 77-s − 79-s − 86-s − 88-s + 98-s − 2·106-s − 2·107-s + ⋯ |
| L(s) = 1 | + 2-s + 7-s − 8-s + 11-s + 14-s − 16-s + 22-s + 23-s + 29-s − 37-s − 43-s + 46-s + 49-s − 2·53-s − 56-s + 58-s + 64-s − 67-s + 71-s − 74-s + 77-s − 79-s − 86-s − 88-s + 98-s − 2·106-s − 2·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.825475986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825475986\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.480324702670257276224191364595, −8.809514046795125395067500520923, −8.116836408344220554470497759149, −6.96970667009939880157183486674, −6.27022701614165737024812194838, −5.22857436838188553221763340267, −4.69504563308156006234533030342, −3.84468175199313457099304344543, −2.88846386044735407266517185951, −1.46488067501830059494637558784,
1.46488067501830059494637558784, 2.88846386044735407266517185951, 3.84468175199313457099304344543, 4.69504563308156006234533030342, 5.22857436838188553221763340267, 6.27022701614165737024812194838, 6.96970667009939880157183486674, 8.116836408344220554470497759149, 8.809514046795125395067500520923, 9.480324702670257276224191364595
