| L(s) = 1 | − 3i·7-s + 2·11-s + 3i·13-s − 6i·17-s − 7·19-s − 6i·23-s − 2·29-s + 5·31-s + 10i·37-s − 12·41-s + 3i·43-s − 10i·47-s − 2·49-s + 6·59-s − 13·61-s + ⋯ |
| L(s) = 1 | − 1.13i·7-s + 0.603·11-s + 0.832i·13-s − 1.45i·17-s − 1.60·19-s − 1.25i·23-s − 0.371·29-s + 0.898·31-s + 1.64i·37-s − 1.87·41-s + 0.457i·43-s − 1.45i·47-s − 0.285·49-s + 0.781·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8542997995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8542997995\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 3iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−8.378352071993944228248165419825, −7.33578740215711270067078119301, −6.69635742473126309489502808889, −6.35380623695166884411936397290, −4.89288333577212779197271176489, −4.47003389671709422926301515525, −3.65349115794712995770391445481, −2.58178692853740257947043494472, −1.47217665067321806933751408304, −0.24567423958343920481522193733,
1.52851275775004625221699349023, 2.36035305996635892209709917576, 3.43356821255530462413775387511, 4.18087893307950134476292047544, 5.23516665801931411758293490787, 5.97306266005699987579140612584, 6.38922597215635954717057164633, 7.50060389476249595159817544329, 8.260165788234170605467004031867, 8.810416026631878741742313844306
