| L(s) = 1 | + i·2-s − 4-s + (−1 + 2i)5-s − i·7-s − i·8-s + (−2 − i)10-s + 3·11-s − 4i·13-s + 14-s + 16-s − 3i·17-s + (1 − 2i)20-s + 3i·22-s + 8i·23-s + (−3 − 4i)25-s + 4·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.377i·7-s − 0.353i·8-s + (−0.632 − 0.316i)10-s + 0.904·11-s − 1.10i·13-s + 0.267·14-s + 0.250·16-s − 0.727i·17-s + (0.223 − 0.447i)20-s + 0.639i·22-s + 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.784·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 15T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
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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
−8.094099213721798193424488645594, −7.52083549456484045938942379210, −7.05240741906682665931247352081, −6.21614037936098522420206413330, −5.50785025972684359313541231204, −4.56578273407024968030960408844, −3.56401479701555360784424164913, −3.12545050524798966065370736241, −1.51572764639780949479465689298, 0,
1.42059860346385328081566522461, 2.13494279949324923490224089553, 3.52244027681771107503263453955, 4.14141363116722610732375571437, 4.80099849602645457782321269118, 5.74221465851506788704358321511, 6.59249709274775176244840498578, 7.48568662731969665417200999178, 8.470420788418262177999678875672
