| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s + 3·13-s − 5·17-s + 19-s + 21-s − 27-s + 6·29-s + 33-s − 5·37-s − 3·39-s + 11·41-s − 8·43-s − 8·47-s + 49-s + 5·51-s − 53-s − 57-s − 10·59-s + 61-s − 63-s − 11·67-s − 9·71-s − 73-s + 77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 1.21·17-s + 0.229·19-s + 0.218·21-s − 0.192·27-s + 1.11·29-s + 0.174·33-s − 0.821·37-s − 0.480·39-s + 1.71·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.700·51-s − 0.137·53-s − 0.132·57-s − 1.30·59-s + 0.128·61-s − 0.125·63-s − 1.34·67-s − 1.06·71-s − 0.117·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4631867319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4631867319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.54913935781842, −12.02079979640175, −11.51109048056293, −11.20649351353062, −10.67946644709615, −10.36684706364737, −9.847792880658704, −9.358364907347023, −8.799375120603171, −8.510449527978171, −7.926847071652311, −7.360829198619116, −6.878468448106305, −6.438356584289355, −6.047276086901308, −5.616414948719347, −4.926956645144874, −4.521397210982948, −4.117169761572208, −3.351819111721400, −2.968906514893783, −2.322561995232264, −1.572522034765638, −1.133941143982448, −0.1913866603276765,
0.1913866603276765, 1.133941143982448, 1.572522034765638, 2.322561995232264, 2.968906514893783, 3.351819111721400, 4.117169761572208, 4.521397210982948, 4.926956645144874, 5.616414948719347, 6.047276086901308, 6.438356584289355, 6.878468448106305, 7.360829198619116, 7.926847071652311, 8.510449527978171, 8.799375120603171, 9.358364907347023, 9.847792880658704, 10.36684706364737, 10.67946644709615, 11.20649351353062, 11.51109048056293, 12.02079979640175, 12.54913935781842
