| L(s) = 1 | − 5.14·2-s + 1.13·3-s + 18.4·4-s − 5·5-s − 5.82·6-s + 26.5·7-s − 53.9·8-s − 25.7·9-s + 25.7·10-s − 65.6·11-s + 20.9·12-s + 44.3·13-s − 136.·14-s − 5.65·15-s + 129.·16-s + 132.·18-s − 37.6·19-s − 92.4·20-s + 30.0·21-s + 337.·22-s − 194.·23-s − 61.0·24-s + 25·25-s − 228.·26-s − 59.6·27-s + 491.·28-s − 157.·29-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.217·3-s + 2.31·4-s − 0.447·5-s − 0.396·6-s + 1.43·7-s − 2.38·8-s − 0.952·9-s + 0.813·10-s − 1.79·11-s + 0.503·12-s + 0.946·13-s − 2.61·14-s − 0.0973·15-s + 2.02·16-s + 1.73·18-s − 0.454·19-s − 1.03·20-s + 0.312·21-s + 3.27·22-s − 1.76·23-s − 0.519·24-s + 0.200·25-s − 1.72·26-s − 0.425·27-s + 3.31·28-s − 1.01·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4695233642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4695233642\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 5.14T + 8T^{2} \) |
| 3 | \( 1 - 1.13T + 27T^{2} \) |
| 7 | \( 1 - 26.5T + 343T^{2} \) |
| 11 | \( 1 + 65.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 37.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 287.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 96.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 106.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 284.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 396.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 573.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 574.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 184.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 626.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 454.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 123.T + 9.12e5T^{2} \) |
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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
−8.864135879688148725719782263144, −8.267477731123925054485611587328, −7.948134727214406978670503739178, −7.34501833804481676801694987849, −5.99372141496074779536730766950, −5.28079708484366986062078897466, −3.78696587022333988761968590695, −2.44130502303954934113749804523, −1.82856799579203534916551958982, −0.42189989955654717481352653970,
0.42189989955654717481352653970, 1.82856799579203534916551958982, 2.44130502303954934113749804523, 3.78696587022333988761968590695, 5.28079708484366986062078897466, 5.99372141496074779536730766950, 7.34501833804481676801694987849, 7.948134727214406978670503739178, 8.267477731123925054485611587328, 8.864135879688148725719782263144
