| L(s) = 1 | + 16i·7-s + 28·11-s − 26i·13-s − 62i·17-s + 68·19-s + 208i·23-s − 58·29-s + 160·31-s − 270i·37-s − 282·41-s + 76i·43-s − 280i·47-s + 87·49-s + 210i·53-s + 196·59-s + ⋯ |
| L(s) = 1 | + 0.863i·7-s + 0.767·11-s − 0.554i·13-s − 0.884i·17-s + 0.821·19-s + 1.88i·23-s − 0.371·29-s + 0.926·31-s − 1.19i·37-s − 1.07·41-s + 0.269i·43-s − 0.868i·47-s + 0.253·49-s + 0.544i·53-s + 0.432·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.321748752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.321748752\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 16iT - 343T^{2} \) |
| 11 | \( 1 - 28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 62iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 68T + 6.85e3T^{2} \) |
| 23 | \( 1 - 208iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 58T + 2.43e4T^{2} \) |
| 31 | \( 1 - 160T + 2.97e4T^{2} \) |
| 37 | \( 1 + 270iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 282T + 6.89e4T^{2} \) |
| 43 | \( 1 - 76iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 280iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 210iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 196T + 2.05e5T^{2} \) |
| 61 | \( 1 - 742T + 2.26e5T^{2} \) |
| 67 | \( 1 + 836iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 504T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.06e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 768T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 726T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.40e3iT - 9.12e5T^{2} \) |
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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
−9.204228913108074211880565604933, −8.173142172806743745609899651278, −7.42494329269320601845727624980, −6.60541615556412237446801323552, −5.56227625481477306846882009236, −5.17160779038040815312207575059, −3.84079336180424578348019153539, −3.05242714601186911184062790348, −1.96500626617487239170112722824, −0.800159099604447128406683906618,
0.68640932598309863670776767711, 1.64364791730528114507182949577, 2.92605587918098656311022051764, 4.01399519323285417596731308778, 4.51365348504950400316155468784, 5.69816088444719818869763823411, 6.70334404603791330287367935369, 7.02179646754100860276078653037, 8.264080731233663780909596707267, 8.658663339604449576886294804016
