| L(s) = 1 | − 17.4·5-s − 4.62·7-s − 11·11-s + 85.1·13-s + 6.08·17-s + 52.0·19-s + 183.·23-s + 179.·25-s − 140.·29-s − 250.·31-s + 80.7·35-s − 203.·37-s + 22.3·41-s − 117.·43-s + 275.·47-s − 321.·49-s − 26.6·53-s + 191.·55-s − 515.·59-s − 693.·61-s − 1.48e3·65-s + 341.·67-s − 831.·71-s + 251.·73-s + 50.8·77-s − 917.·79-s + 456.·83-s + ⋯ |
| L(s) = 1 | − 1.56·5-s − 0.249·7-s − 0.301·11-s + 1.81·13-s + 0.0868·17-s + 0.628·19-s + 1.66·23-s + 1.43·25-s − 0.899·29-s − 1.45·31-s + 0.389·35-s − 0.904·37-s + 0.0850·41-s − 0.416·43-s + 0.855·47-s − 0.937·49-s − 0.0690·53-s + 0.470·55-s − 1.13·59-s − 1.45·61-s − 2.83·65-s + 0.622·67-s − 1.38·71-s + 0.403·73-s + 0.0753·77-s − 1.30·79-s + 0.604·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 17.4T + 125T^{2} \) |
| 7 | \( 1 + 4.62T + 343T^{2} \) |
| 13 | \( 1 - 85.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.08T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 203.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 117.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 26.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 515.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 341.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 831.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 917.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 456.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 91.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 146.T + 9.12e5T^{2} \) |
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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
−9.194163953758969431321921281487, −8.637958016946315407584349161818, −7.67764411989033540791499179387, −7.08828076431629336348224554504, −5.93514562057468887372356553418, −4.84988348078150424514042191228, −3.68739411328846875572343947726, −3.24092454744599369339478430074, −1.29320725804544686150463237272, 0,
1.29320725804544686150463237272, 3.24092454744599369339478430074, 3.68739411328846875572343947726, 4.84988348078150424514042191228, 5.93514562057468887372356553418, 7.08828076431629336348224554504, 7.67764411989033540791499179387, 8.637958016946315407584349161818, 9.194163953758969431321921281487
