| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s + 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s + 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 439 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 439 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.710038164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.710038164\) |
| \(L(1)\) |
\(\approx\) |
\(2.249100549\) |
| \(L(1)\) |
\(\approx\) |
\(2.249100549\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 439 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.88760287994976805314338724500, −22.870604736477729818819853480, −22.07495860126388523724797103880, −21.61905773442278832206231701681, −20.73624340387215040734828494676, −19.92457950024479518041816709289, −18.319339222932588146934870109330, −17.760700131787352216450590867420, −16.865410023507102330115034244687, −16.051737101051494663220281666891, −15.052667685084190826885272001130, −13.95641216365738389191394078947, −13.51172450886717688658901620313, −12.29883777056091124326951615393, −11.54109930468909751098155281011, −10.86314427374048950010855624410, −9.90845457981524834712354769397, −8.516295457721022433194954552107, −7.02325643684754003556513098976, −6.33422637511388924939278286284, −5.47738689296295486093531597960, −4.71237926756550747370751744630, −3.65960625020430255955394903674, −1.91372463658556351550855263337, −1.28279077144730586854866903783,
1.28279077144730586854866903783, 1.91372463658556351550855263337, 3.65960625020430255955394903674, 4.71237926756550747370751744630, 5.47738689296295486093531597960, 6.33422637511388924939278286284, 7.02325643684754003556513098976, 8.516295457721022433194954552107, 9.90845457981524834712354769397, 10.86314427374048950010855624410, 11.54109930468909751098155281011, 12.29883777056091124326951615393, 13.51172450886717688658901620313, 13.95641216365738389191394078947, 15.052667685084190826885272001130, 16.051737101051494663220281666891, 16.865410023507102330115034244687, 17.760700131787352216450590867420, 18.319339222932588146934870109330, 19.92457950024479518041816709289, 20.73624340387215040734828494676, 21.61905773442278832206231701681, 22.07495860126388523724797103880, 22.870604736477729818819853480, 23.88760287994976805314338724500
