| L(s) = 1 | + 8.57·3-s − 8.24·5-s + 26.8·7-s + 46.5·9-s + 11·11-s − 72.1·13-s − 70.7·15-s + 116.·17-s − 70.6·19-s + 230.·21-s + 21.2·23-s − 57.0·25-s + 167.·27-s + 38.4·29-s − 263.·31-s + 94.3·33-s − 221.·35-s − 156.·37-s − 618.·39-s − 112.·41-s + 59.7·43-s − 383.·45-s − 134.·47-s + 376.·49-s + 1.00e3·51-s − 585.·53-s − 90.6·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s − 0.737·5-s + 1.44·7-s + 1.72·9-s + 0.301·11-s − 1.53·13-s − 1.21·15-s + 1.66·17-s − 0.852·19-s + 2.39·21-s + 0.192·23-s − 0.456·25-s + 1.19·27-s + 0.246·29-s − 1.52·31-s + 0.497·33-s − 1.06·35-s − 0.694·37-s − 2.53·39-s − 0.426·41-s + 0.211·43-s − 1.27·45-s − 0.417·47-s + 1.09·49-s + 2.74·51-s − 1.51·53-s − 0.222·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.453770397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453770397\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 8.57T + 27T^{2} \) |
| 5 | \( 1 + 8.24T + 125T^{2} \) |
| 7 | \( 1 - 26.8T + 343T^{2} \) |
| 13 | \( 1 + 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 21.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 38.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 112.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 59.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 585.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 573.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 292.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 230.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 763.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 832.T + 9.12e5T^{2} \) |
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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
−14.15112838371468489555260026407, −12.64555432174283018800162517979, −11.67000859361383516785327882692, −10.19190512681691691654648637078, −8.982482989224976186176405820413, −7.910315789992655858607359498043, −7.47839850254553866370060509211, −4.88537701709424161140493738813, −3.56759448675284572070662362310, −1.95346657413988652683036824863,
1.95346657413988652683036824863, 3.56759448675284572070662362310, 4.88537701709424161140493738813, 7.47839850254553866370060509211, 7.910315789992655858607359498043, 8.982482989224976186176405820413, 10.19190512681691691654648637078, 11.67000859361383516785327882692, 12.64555432174283018800162517979, 14.15112838371468489555260026407
