| L(s) = 1 | + i·3-s − i·5-s − 4.68·7-s − 9-s − 2.29i·11-s − 4.97i·13-s + 15-s − 2.97·17-s − 2.68i·19-s − 4.68i·21-s − 2.68·23-s − 25-s − i·27-s + 2i·29-s + 6.97·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 1.77·7-s − 0.333·9-s − 0.691i·11-s − 1.38i·13-s + 0.258·15-s − 0.722·17-s − 0.616i·19-s − 1.02i·21-s − 0.560·23-s − 0.200·25-s − 0.192i·27-s + 0.371i·29-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.277095 - 0.464334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.277095 - 0.464334i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 + 2.29iT - 11T^{2} \) |
| 13 | \( 1 + 4.97iT - 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 2.68iT - 19T^{2} \) |
| 23 | \( 1 + 2.68T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 - 4.39iT - 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 9.37iT - 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 1.70iT - 59T^{2} \) |
| 61 | \( 1 - 4.58iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 0.585T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 - 13.3iT - 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 3.95T + 97T^{2} \) |
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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
−10.37907892712245691631589318761, −9.964670976261923512075273488451, −8.950665628101220495679353764531, −8.256937188863111744350471035141, −6.81675580218907457479090101108, −6.00319611029417306373406977669, −5.00054450372563706682042124835, −3.63477127794108134686480423389, −2.86469834856782493836041934656, −0.30777006703300730784593818120,
2.04794629328115188020690145350, 3.25991723543608920073604704656, 4.43096201317840401233505175560, 6.17783426882993611343289390709, 6.56930010474206238332695508003, 7.39084872249469296557237697923, 8.664910071832801087706570949208, 9.656900934031127957986118156355, 10.15265002605815629689099595589, 11.47785047901264266710102779233
