L(s) = 1 | − 0.189i·3-s + (−1.60 − 1.55i)5-s − 1.65i·7-s + 2.96·9-s − 0.0759·11-s − 1.24i·13-s + (−0.295 + 0.303i)15-s − 5.17i·17-s + 0.792·19-s − 0.313·21-s + i·23-s + (0.137 + 4.99i)25-s − 1.12i·27-s + 2.85·29-s − 8.90·31-s + ⋯ |
L(s) = 1 | − 0.109i·3-s + (−0.716 − 0.697i)5-s − 0.626i·7-s + 0.988·9-s − 0.0228·11-s − 0.344i·13-s + (−0.0762 + 0.0783i)15-s − 1.25i·17-s + 0.181·19-s − 0.0684·21-s + 0.208i·23-s + (0.0275 + 0.999i)25-s − 0.217i·27-s + 0.529·29-s − 1.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.183972477\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183972477\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.60 + 1.55i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 0.189iT - 3T^{2} \) |
| 7 | \( 1 + 1.65iT - 7T^{2} \) |
| 11 | \( 1 + 0.0759T + 11T^{2} \) |
| 13 | \( 1 + 1.24iT - 13T^{2} \) |
| 17 | \( 1 + 5.17iT - 17T^{2} \) |
| 19 | \( 1 - 0.792T + 19T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 - 1.75iT - 43T^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 6.96iT - 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 + 7.49iT - 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 6.68iT - 83T^{2} \) |
| 89 | \( 1 - 3.03T + 89T^{2} \) |
| 97 | \( 1 - 3.21iT - 97T^{2} \) |
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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
−9.175271917722272201024855224727, −7.87864308374849594833442349499, −7.53663553010286731572004905441, −6.82730312748059196530317221922, −5.59696576533926234973342825145, −4.71400528035000014650357312922, −4.08305732267150395481710354715, −3.10360054793550769421239122317, −1.56732596191691130658639781153, −0.45735994679497252389361585157,
1.55928326413005026797748177561, 2.73930032432619278230467961537, 3.80312110057999429344745439455, 4.41195484093979967977737084738, 5.58129074710336355546221006549, 6.47660187240621716266469831456, 7.17798119575262470289929018403, 7.926638999255609411390645417321, 8.731511964844073721931211065084, 9.531239008163878588796960554133