| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 13-s + 17-s + 19-s + 21-s − 23-s + 27-s − 29-s + 31-s − 33-s + 39-s + 41-s − 43-s + 47-s + 49-s + 51-s − 53-s + 57-s + 59-s − 61-s + 63-s + 67-s − 69-s − 71-s + ⋯ |
| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s + 13-s + 17-s + 19-s + 21-s − 23-s + 27-s − 29-s + 31-s − 33-s + 39-s + 41-s − 43-s + 47-s + 49-s + 51-s − 53-s + 57-s + 59-s − 61-s + 63-s + 67-s − 69-s − 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.024828474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.024828474\) |
| \(L(1)\) |
\(\approx\) |
\(1.847795884\) |
| \(L(1)\) |
\(\approx\) |
\(1.847795884\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 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
−22.10804340502595935379554231123, −21.11501261602350066424459549324, −20.75330403800363487594316085430, −20.10765219690657550802013353644, −18.892443841273588638263793659068, −18.39452137300464468934155810046, −17.675615537164991943728238642560, −16.33601432837091218967262439638, −15.68256188058293782948980387684, −14.857193260390856817510099094657, −14.000938028270049419325970631785, −13.509885475750767114926773380353, −12.465793742121436646355292547892, −11.48262465214826261683201770409, −10.498988939174473576185374591470, −9.72270893605356534504178189151, −8.65609584924049958060732861309, −7.90952109231711077770895871241, −7.4584815401961153755703104182, −5.96004625420480685413151656039, −5.01255761355168080389425410429, −3.95432358287188581693804662797, −3.04239274097160140685190621316, −1.97390147237529426831933216849, −1.01811297534319381391507871328,
1.01811297534319381391507871328, 1.97390147237529426831933216849, 3.04239274097160140685190621316, 3.95432358287188581693804662797, 5.01255761355168080389425410429, 5.96004625420480685413151656039, 7.4584815401961153755703104182, 7.90952109231711077770895871241, 8.65609584924049958060732861309, 9.72270893605356534504178189151, 10.498988939174473576185374591470, 11.48262465214826261683201770409, 12.465793742121436646355292547892, 13.509885475750767114926773380353, 14.000938028270049419325970631785, 14.857193260390856817510099094657, 15.68256188058293782948980387684, 16.33601432837091218967262439638, 17.675615537164991943728238642560, 18.39452137300464468934155810046, 18.892443841273588638263793659068, 20.10765219690657550802013353644, 20.75330403800363487594316085430, 21.11501261602350066424459549324, 22.10804340502595935379554231123
