| L(s) = 1 | + (−1.70 − 0.323i)3-s − 1.55i·5-s + (2.79 + 1.09i)9-s − 2.53·13-s + (−0.502 + 2.64i)15-s − 4.33i·17-s + 6.83i·19-s + 3.80·23-s + 2.58·25-s + (−4.39 − 2.77i)27-s − 8.33i·29-s − 4.89i·31-s + 1.58·37-s + (4.31 + 0.818i)39-s − 7.44i·41-s + ⋯ |
| L(s) = 1 | + (−0.982 − 0.186i)3-s − 0.695i·5-s + (0.930 + 0.366i)9-s − 0.702·13-s + (−0.129 + 0.683i)15-s − 1.05i·17-s + 1.56i·19-s + 0.794·23-s + 0.516·25-s + (−0.845 − 0.533i)27-s − 1.54i·29-s − 0.879i·31-s + 0.260·37-s + (0.690 + 0.131i)39-s − 1.16i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7977391399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7977391399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 + 0.323i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.55iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + 4.33iT - 17T^{2} \) |
| 19 | \( 1 - 6.83iT - 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 + 8.33iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 + 7.44iT - 41T^{2} \) |
| 43 | \( 1 - 6.20iT - 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 + 4.33iT - 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−8.762214982012456144603345586614, −7.65963006046675124786211055174, −7.32469088479398256713862997019, −6.14962243397429904731129016520, −5.63341971771728375033501794682, −4.73622979067894349241907410656, −4.17944972025892687961692712390, −2.72057258673836286842651512210, −1.47500823857382520322200384859, −0.35053350011636258480582990773,
1.20942598145649040012859361967, 2.62621740154058830251403806152, 3.57629730634821820345350084207, 4.72919472374402745914503222513, 5.19862510894377594576658382922, 6.27518700268741614861439763676, 6.92381767257114072811621096232, 7.34664916072588469092078344261, 8.645357549483966616725269180235, 9.284978816830397613448748811799
