L(s) = 1 | − i·2-s − 4-s + 4.23·5-s + (2.64 − 0.145i)7-s + i·8-s − 4.23i·10-s − 0.875i·11-s + 3.49i·13-s + (−0.145 − 2.64i)14-s + 16-s + 0.180·17-s − i·19-s − 4.23·20-s − 0.875·22-s + 1.13i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.89·5-s + (0.998 − 0.0549i)7-s + 0.353i·8-s − 1.33i·10-s − 0.264i·11-s + 0.970i·13-s + (−0.0388 − 0.706i)14-s + 0.250·16-s + 0.0437·17-s − 0.229i·19-s − 0.946·20-s − 0.186·22-s + 0.237i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.836190570\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.836190570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.64 + 0.145i)T \) |
| 19 | \( 1 + iT \) |
good | 5 | \( 1 - 4.23T + 5T^{2} \) |
| 11 | \( 1 + 0.875iT - 11T^{2} \) |
| 13 | \( 1 - 3.49iT - 13T^{2} \) |
| 17 | \( 1 - 0.180T + 17T^{2} \) |
| 23 | \( 1 - 1.13iT - 23T^{2} \) |
| 29 | \( 1 + 9.84iT - 29T^{2} \) |
| 31 | \( 1 + 1.21iT - 31T^{2} \) |
| 37 | \( 1 + 6.29T + 37T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 - 0.254T + 43T^{2} \) |
| 47 | \( 1 + 8.88T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 6.24iT - 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 - 3.26iT - 71T^{2} \) |
| 73 | \( 1 - 16.2iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 1.87T + 89T^{2} \) |
| 97 | \( 1 + 6.10iT - 97T^{2} \) |
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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
−9.030637002040069153260342306320, −8.400768472187663526044883255930, −7.31634160967802930785193281671, −6.30783070348024428286024080625, −5.65703937256916245824372836623, −4.88929021433077389919796781910, −4.06747630652854312042727365732, −2.63397936810007466391134907221, −2.01239705274977261161326457109, −1.18527545536550587898992647838,
1.24236342253136298369036690742, 2.12549261058558133543199312083, 3.27619846883384820413340624261, 4.76994774865087744727876517764, 5.26558462794900114690071442501, 5.85161064285235995126250249400, 6.69572444699282980406609546590, 7.44418535351852112182137122358, 8.438609474796705216005544489920, 8.920471989069147563114056032297