| L(s) = 1 | − 3.12i·3-s − i·5-s + 2.84·7-s − 6.78·9-s − 1.16·11-s − 3.10·13-s − 3.12·15-s + 4.10i·17-s + 2.04·19-s − 8.91i·21-s + (−4.30 + 2.10i)23-s − 25-s + 11.8i·27-s − 6.42·29-s − 1.91i·31-s + ⋯ |
| L(s) = 1 | − 1.80i·3-s − 0.447i·5-s + 1.07·7-s − 2.26·9-s − 0.351·11-s − 0.859·13-s − 0.807·15-s + 0.996i·17-s + 0.468·19-s − 1.94i·21-s + (−0.898 + 0.439i)23-s − 0.200·25-s + 2.28i·27-s − 1.19·29-s − 0.343i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06359178102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06359178102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (4.30 - 2.10i)T \) |
| good | 3 | \( 1 + 3.12iT - 3T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 - 4.10iT - 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 29 | \( 1 + 6.42T + 29T^{2} \) |
| 31 | \( 1 + 1.91iT - 31T^{2} \) |
| 37 | \( 1 - 0.291iT - 37T^{2} \) |
| 41 | \( 1 + 2.90T + 41T^{2} \) |
| 43 | \( 1 + 1.51T + 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 3.23iT - 53T^{2} \) |
| 59 | \( 1 + 1.04iT - 59T^{2} \) |
| 61 | \( 1 + 7.29iT - 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 - 12.0iT - 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 4.77T + 83T^{2} \) |
| 89 | \( 1 - 6.54iT - 89T^{2} \) |
| 97 | \( 1 + 18.3iT - 97T^{2} \) |
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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
−8.318643844889240282249949990350, −7.76824365531244877109267017717, −7.52055758029866043095699805552, −6.52105813178577223819656393352, −5.73736407900468158661217363771, −5.17704783387941011975273377672, −4.10695290408169140947726005656, −2.80365260019772813280430755027, −1.86508291288616019256347339859, −1.37602893214570691057976596085,
0.01744718326204588425089085354, 2.08833184144931814815508260898, 2.99398819762457305360097784566, 3.82777452317221135781724104547, 4.66466505777409097153949992831, 5.14863333585793117517764730156, 5.75776444231759912917423766624, 7.01107674840385082645261530035, 7.78979726209974185515618519991, 8.503486385322221804741825343558
