| L(s) = 1 | + (−1 − i)3-s + (−3 + 3i)7-s − i·9-s − 2i·11-s + (3 − 3i)13-s + (−1 − i)17-s + 4·19-s + 6·21-s + (−1 − i)23-s + (−4 + 4i)27-s + 10i·31-s + (−2 + 2i)33-s + (−1 − i)37-s − 6·39-s − 10·41-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.577i)3-s + (−1.13 + 1.13i)7-s − 0.333i·9-s − 0.603i·11-s + (0.832 − 0.832i)13-s + (−0.242 − 0.242i)17-s + 0.917·19-s + 1.30·21-s + (−0.208 − 0.208i)23-s + (−0.769 + 0.769i)27-s + 1.79i·31-s + (−0.348 + 0.348i)33-s + (−0.164 − 0.164i)37-s − 0.960·39-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (1 + i)T + 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + (5 + 5i)T + 43iT^{2} \) |
| 47 | \( 1 + (3 - 3i)T - 47iT^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (-1 + i)T - 67iT^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + (1 - i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (5 + 5i)T + 83iT^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−8.912927045451480904730248988512, −8.283539223919954088699060381263, −7.08197431740022071042206041486, −6.40693536831449581400013557028, −5.81728526750977657581662811718, −5.14296477111565930907404971565, −3.43677035799624185483612676385, −3.00867318863603356950809490039, −1.38134933259869946030630000976, 0,
1.68082192265665586254738887024, 3.26524033774252583616087916396, 4.08010266609314751459190029639, 4.76596156559381746265679850396, 5.89290166053203571505447211701, 6.60444173453214833013871216286, 7.35466130558282302581545567147, 8.238087443664100648297461376930, 9.492720387548640946458888795814
