| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 5·11-s − 13-s + 15-s − 5·17-s − 2·19-s − 2·21-s + 4·23-s + 25-s − 27-s + 29-s − 7·31-s + 5·33-s − 2·35-s − 2·37-s + 39-s + 6·41-s − 43-s − 45-s − 10·47-s − 3·49-s + 5·51-s − 9·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 1.21·17-s − 0.458·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s − 1.25·31-s + 0.870·33-s − 0.338·35-s − 0.328·37-s + 0.160·39-s + 0.937·41-s − 0.152·43-s − 0.149·45-s − 1.45·47-s − 3/7·49-s + 0.700·51-s − 1.23·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.92265715408199, −12.59398305145977, −12.10957491607784, −11.35721700716930, −11.12531737370001, −10.89373489495097, −10.50981094886998, −9.769592636573398, −9.445411483549675, −8.769321275361794, −8.362194408711312, −7.801925697420669, −7.650877052167142, −6.873033035045734, −6.631117081049072, −5.993291282080086, −5.284507080655491, −4.961973366593534, −4.772958125291598, −4.038152248172048, −3.528060368628673, −2.778032619151324, −2.306296137906730, −1.755090738097585, −1.030239714516477, 0, 0,
1.030239714516477, 1.755090738097585, 2.306296137906730, 2.778032619151324, 3.528060368628673, 4.038152248172048, 4.772958125291598, 4.961973366593534, 5.284507080655491, 5.993291282080086, 6.631117081049072, 6.873033035045734, 7.650877052167142, 7.801925697420669, 8.362194408711312, 8.769321275361794, 9.445411483549675, 9.769592636573398, 10.50981094886998, 10.89373489495097, 11.12531737370001, 11.35721700716930, 12.10957491607784, 12.59398305145977, 12.92265715408199
