| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s − 13-s + 15-s − 17-s + 19-s + 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s + 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s + 53-s + 55-s − 57-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 11-s − 13-s + 15-s − 17-s + 19-s + 21-s + 23-s + 25-s − 27-s − 29-s + 31-s + 33-s + 35-s + 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s + 53-s + 55-s − 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5090392253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5090392253\) |
| \(L(1)\) |
\(\approx\) |
\(0.5164746507\) |
| \(L(1)\) |
\(\approx\) |
\(0.5164746507\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 \) |
| good | 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 | \( 1 + T \) |
| 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
−28.055929388343378262423651035122, −26.86259990788137381775504418031, −26.33956052661445138027422280555, −24.62117613901229929362366241268, −23.9151107348469192225696089456, −22.72266633666223580699904520275, −22.48803894206718246379167085679, −21.07172436882948145435110476809, −19.759548636801391083349159962029, −18.94345085355237349573807131880, −17.90185163118126319447330078031, −16.68778427474795209157130932474, −15.88460037020232438567587937272, −15.16233624302250647841480000421, −13.25896422315704832833979452990, −12.47909993027605684188687266731, −11.47446809121786589720706013463, −10.4851387649294303925375890204, −9.33777644469170663419961742082, −7.6207396362520879513361003277, −6.840412086404635360259461624026, −5.422004887766601355090790590072, −4.331561162560183161429336269974, −2.84198705815088156092677036033, −0.50489835112590698642863946765,
0.50489835112590698642863946765, 2.84198705815088156092677036033, 4.331561162560183161429336269974, 5.422004887766601355090790590072, 6.840412086404635360259461624026, 7.6207396362520879513361003277, 9.33777644469170663419961742082, 10.4851387649294303925375890204, 11.47446809121786589720706013463, 12.47909993027605684188687266731, 13.25896422315704832833979452990, 15.16233624302250647841480000421, 15.88460037020232438567587937272, 16.68778427474795209157130932474, 17.90185163118126319447330078031, 18.94345085355237349573807131880, 19.759548636801391083349159962029, 21.07172436882948145435110476809, 22.48803894206718246379167085679, 22.72266633666223580699904520275, 23.9151107348469192225696089456, 24.62117613901229929362366241268, 26.33956052661445138027422280555, 26.86259990788137381775504418031, 28.055929388343378262423651035122
