| L(s) = 1 | − 7.72·3-s − 18.6·5-s + 28.1·7-s + 32.7·9-s + 44.4·11-s + 67.2·13-s + 143.·15-s − 21.2·17-s + 151.·19-s − 217.·21-s − 23·23-s + 221.·25-s − 44.0·27-s + 164.·29-s + 120.·31-s − 343.·33-s − 524.·35-s + 188.·37-s − 519.·39-s − 59.5·41-s − 432.·43-s − 608.·45-s + 555.·47-s + 451.·49-s + 164.·51-s + 106.·53-s − 827.·55-s + ⋯ |
| L(s) = 1 | − 1.48·3-s − 1.66·5-s + 1.52·7-s + 1.21·9-s + 1.21·11-s + 1.43·13-s + 2.47·15-s − 0.303·17-s + 1.82·19-s − 2.26·21-s − 0.208·23-s + 1.77·25-s − 0.314·27-s + 1.05·29-s + 0.700·31-s − 1.81·33-s − 2.53·35-s + 0.839·37-s − 2.13·39-s − 0.226·41-s − 1.53·43-s − 2.01·45-s + 1.72·47-s + 1.31·49-s + 0.450·51-s + 0.276·53-s − 2.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.476178219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476178219\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 7.72T + 27T^{2} \) |
| 5 | \( 1 + 18.6T + 125T^{2} \) |
| 7 | \( 1 - 28.1T + 343T^{2} \) |
| 11 | \( 1 - 44.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 21.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 151.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 59.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 432.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 555.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 571.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 37.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 94.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 6.85T + 3.57e5T^{2} \) |
| 73 | \( 1 - 194.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 12.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 682.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−8.908716730556660381906235141878, −8.237425085007902047080101327433, −7.49958533632258940379596541152, −6.68529149947993891720702869995, −5.82203570554423947594225853644, −4.81091255969224530019384654138, −4.31959279849179437354974343203, −3.40605705360710064514985473825, −1.27723077123145478346334681322, −0.798678382865003062382215297000,
0.798678382865003062382215297000, 1.27723077123145478346334681322, 3.40605705360710064514985473825, 4.31959279849179437354974343203, 4.81091255969224530019384654138, 5.82203570554423947594225853644, 6.68529149947993891720702869995, 7.49958533632258940379596541152, 8.237425085007902047080101327433, 8.908716730556660381906235141878
