| L(s) = 1 | + 3.41·5-s − 7-s + 2.82·11-s + 5.65·13-s − 2.24·17-s + 19-s + 8.48·23-s + 6.65·25-s − 10.2·29-s + 8.82·31-s − 3.41·35-s − 6·37-s + 8.82·41-s − 8.82·43-s + 10.5·47-s + 49-s + 5.07·53-s + 9.65·55-s + 2.34·59-s + 2·61-s + 19.3·65-s − 9.17·67-s − 12.7·71-s − 8.82·73-s − 2.82·77-s + 2.34·79-s + 12.2·83-s + ⋯ |
| L(s) = 1 | + 1.52·5-s − 0.377·7-s + 0.852·11-s + 1.56·13-s − 0.543·17-s + 0.229·19-s + 1.76·23-s + 1.33·25-s − 1.90·29-s + 1.58·31-s − 0.577·35-s − 0.986·37-s + 1.37·41-s − 1.34·43-s + 1.54·47-s + 0.142·49-s + 0.696·53-s + 1.30·55-s + 0.305·59-s + 0.256·61-s + 2.39·65-s − 1.12·67-s − 1.51·71-s − 1.03·73-s − 0.322·77-s + 0.263·79-s + 1.34·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.500442986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.500442986\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 9.17T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−7.54694947114195371992072396844, −6.78634239099312929708631910917, −6.28853273802384896512902423654, −5.78461370774411260812233120190, −5.10996051493200863459940093253, −4.13950735315767457034533560760, −3.38850080721243033707546338320, −2.56091492195615597908708683710, −1.62583553750353325593138593093, −0.982755894106937492270529989039,
0.982755894106937492270529989039, 1.62583553750353325593138593093, 2.56091492195615597908708683710, 3.38850080721243033707546338320, 4.13950735315767457034533560760, 5.10996051493200863459940093253, 5.78461370774411260812233120190, 6.28853273802384896512902423654, 6.78634239099312929708631910917, 7.54694947114195371992072396844
