| L(s) = 1 | − 2.28i·3-s − 0.540·5-s + (−1.41 − 2.23i)7-s − 2.23·9-s + 1.23·11-s + 2.28·13-s + 1.23i·15-s − 1.74i·17-s − 5.11i·19-s + (−5.11 + 3.23i)21-s − 3.23i·23-s − 4.70·25-s − 1.74i·27-s + 2i·29-s − 3.90·31-s + ⋯ |
| L(s) = 1 | − 1.32i·3-s − 0.241·5-s + (−0.534 − 0.845i)7-s − 0.745·9-s + 0.372·11-s + 0.634·13-s + 0.319i·15-s − 0.423i·17-s − 1.17i·19-s + (−1.11 + 0.706i)21-s − 0.674i·23-s − 0.941·25-s − 0.336i·27-s + 0.371i·29-s − 0.702·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067925132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067925132\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
| good | 3 | \( 1 + 2.28iT - 3T^{2} \) |
| 5 | \( 1 + 0.540T + 5T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 1.74iT - 17T^{2} \) |
| 19 | \( 1 + 5.11iT - 19T^{2} \) |
| 23 | \( 1 + 3.23iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 - 6.94iT - 37T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 8.47iT - 53T^{2} \) |
| 59 | \( 1 + 1.62iT - 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 3.52iT - 79T^{2} \) |
| 83 | \( 1 - 9.02iT - 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 1.74iT - 97T^{2} \) |
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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
−8.791595257101703146839487887614, −7.79226672656627903659270702569, −7.27188655784558365243408507996, −6.61018139148785573636167508929, −5.98810957205601602981969456382, −4.66674607828462949473172656140, −3.72693341975072553891185994495, −2.66964876731016024832518495322, −1.43833525477864567947238136382, −0.40629533843496138244258117427,
1.80332418455986277362875748938, 3.28664061341233643649506078162, 3.78340719133862026100434034225, 4.66022180042443674252792643957, 5.79607876591579386872311400953, 6.09724812837937874394646414446, 7.47109543282448158652670511018, 8.332565150353689140166019990789, 9.136304887255874045307736900415, 9.644518846479087017659094672429
