| L(s) = 1 | − i·3-s + i·5-s + 2.13·7-s − 9-s − 4.16·11-s + 2.81·13-s + 15-s − 1.82i·17-s + 1.32·19-s − 2.13i·21-s + (−4.63 + 1.24i)23-s − 25-s + i·27-s − 4.92·29-s + 10.5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s + 0.806·7-s − 0.333·9-s − 1.25·11-s + 0.781·13-s + 0.258·15-s − 0.441i·17-s + 0.304·19-s − 0.465i·21-s + (−0.965 + 0.259i)23-s − 0.200·25-s + 0.192i·27-s − 0.914·29-s + 1.89i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.349214882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349214882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (4.63 - 1.24i)T \) |
| good | 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 + 1.82iT - 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 10.5iT - 31T^{2} \) |
| 37 | \( 1 + 4.95iT - 37T^{2} \) |
| 41 | \( 1 - 0.656T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 - 7.28iT - 47T^{2} \) |
| 53 | \( 1 - 7.24iT - 53T^{2} \) |
| 59 | \( 1 - 8.42iT - 59T^{2} \) |
| 61 | \( 1 + 3.54iT - 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 0.897iT - 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 + 7.73T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 + 5.21iT - 89T^{2} \) |
| 97 | \( 1 + 0.00964iT - 97T^{2} \) |
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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
−8.101835676639054950416919639928, −7.61047907774426808795935108748, −7.07904169256944888309331572276, −6.06727220502459597237719859620, −5.54004515674408803624560906561, −4.77578548459061251857777340604, −3.76631703869913370635002087011, −2.86637177355949772017409491397, −2.08546071717019203003569702810, −1.11011697299740963322408543388,
0.36604008975109103998308375904, 1.74025356221498873033560504330, 2.56533069955479907958464145440, 3.75360510356695359206557527907, 4.26232783984341089114488650590, 5.21504541380530641781440074562, 5.59477025945643220284307909207, 6.42426735105073904291021664739, 7.60689152624089617857462781076, 8.106415617779783984733415449146
