| L(s) = 1 | − 3-s − 5-s + 9-s − 11-s − 3·13-s + 15-s − 7·19-s + 6·23-s − 4·25-s − 27-s − 9·29-s + 33-s − 3·37-s + 3·39-s − 8·41-s + 10·43-s − 45-s − 3·47-s + 6·53-s + 55-s + 7·57-s − 7·59-s − 10·61-s + 3·65-s − 3·67-s − 6·69-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.258·15-s − 1.60·19-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 1.67·29-s + 0.174·33-s − 0.493·37-s + 0.480·39-s − 1.24·41-s + 1.52·43-s − 0.149·45-s − 0.437·47-s + 0.824·53-s + 0.134·55-s + 0.927·57-s − 0.911·59-s − 1.28·61-s + 0.372·65-s − 0.366·67-s − 0.722·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5639790586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5639790586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.43930976584398, −15.56998013033664, −15.13695314172755, −14.86337698933237, −14.00450191829866, −13.26410280358164, −12.83840549778349, −12.27600394544468, −11.73399108267459, −11.06670046486879, −10.64591015051279, −10.07827326563690, −9.219557131316912, −8.845973879763006, −7.853562879480034, −7.496877148449494, −6.796578304369825, −6.138726515347827, −5.412894991269635, −4.796704360938953, −4.147467064698083, −3.410473939354780, −2.424883363755502, −1.665471040246942, −0.3376533007152822,
0.3376533007152822, 1.665471040246942, 2.424883363755502, 3.410473939354780, 4.147467064698083, 4.796704360938953, 5.412894991269635, 6.138726515347827, 6.796578304369825, 7.496877148449494, 7.853562879480034, 8.845973879763006, 9.219557131316912, 10.07827326563690, 10.64591015051279, 11.06670046486879, 11.73399108267459, 12.27600394544468, 12.83840549778349, 13.26410280358164, 14.00450191829866, 14.86337698933237, 15.13695314172755, 15.56998013033664, 16.43930976584398
