| L(s) = 1 | − 2i·5-s + (2.23 − 1.41i)7-s − 1.41·11-s − 4.47·17-s − 6.32·19-s + 3.16i·23-s + 25-s − 3.16·29-s + 5.65i·31-s + (−2.82 − 4.47i)35-s + 4.47i·37-s − 4.47·41-s − 4i·43-s − 8·47-s + (3.00 − 6.32i)49-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s + (0.845 − 0.534i)7-s − 0.426·11-s − 1.08·17-s − 1.45·19-s + 0.659i·23-s + 0.200·25-s − 0.587·29-s + 1.01i·31-s + (−0.478 − 0.755i)35-s + 0.735i·37-s − 0.698·41-s − 0.609i·43-s − 1.16·47-s + (0.428 − 0.903i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1487223485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1487223485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 3.16iT - 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 9.48iT - 71T^{2} \) |
| 73 | \( 1 - 6.32iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 6.32iT - 97T^{2} \) |
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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
−8.531349226261690544949950424621, −8.227708353583673744366987976100, −7.26346637925103883241447645171, −6.59975754169194201995694045779, −5.61137716263998927273986957609, −4.74518106898512751632281644316, −4.47620060728453223298143094635, −3.38413847533821488868943838182, −2.11129268161478995381942462962, −1.33470196896476974794399842610,
0.03971361901561717906303997567, 1.92235177518961962412563699462, 2.41663990797842577958045699903, 3.45554915536301805792665951306, 4.51327228316132304912203740739, 5.03237650694545094677442197212, 6.30712427113588603257416355493, 6.42336141178300776034941725545, 7.57912396096120214279126613184, 8.092333853928168092025241283122
