L(s) = 1 | − 2i·2-s − 2·4-s − 3i·7-s + 5·11-s − i·13-s − 6·14-s − 4·16-s − 5i·17-s − 2·19-s − 10i·22-s − i·23-s − 2·26-s + 6i·28-s + 10·29-s − 2·31-s + 8i·32-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 4-s − 1.13i·7-s + 1.50·11-s − 0.277i·13-s − 1.60·14-s − 16-s − 1.21i·17-s − 0.458·19-s − 2.13i·22-s − 0.208i·23-s − 0.392·26-s + 1.13i·28-s + 1.85·29-s − 0.359·31-s + 1.41i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.810665242\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.810665242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 + 9iT - 97T^{2} \) |
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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
−8.675566139624296525773197682312, −7.49686593032851406998557892223, −6.88118108496805208111348224235, −6.12942240015596643193643312437, −4.67030000607447335102972201670, −4.22748431481126597095443123280, −3.40357829694951989747006121101, −2.56204718516906521627184556147, −1.34851762432787253956217612460, −0.62469013096305254382519254061,
1.53472755317651973535059309969, 2.66787646879889425397448257677, 3.99007128453281432375083029977, 4.69536624586676876610615779475, 5.71290059337612206925382099007, 6.27135319440274831991477037726, 6.66144753550010207973464178678, 7.66492130684733042809321401829, 8.452786135679513960131907016705, 8.884456166963231014604933930854