| L(s) = 1 | − i·3-s − i·5-s − 4.84·7-s − 9-s − 3.09·11-s − 4.53·13-s − 15-s − 1.04i·17-s − 3.05·19-s + 4.84i·21-s + (−4.79 + 0.0971i)23-s − 25-s + i·27-s − 2.44·29-s + 3.08i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 1.83·7-s − 0.333·9-s − 0.933·11-s − 1.25·13-s − 0.258·15-s − 0.254i·17-s − 0.701·19-s + 1.05i·21-s + (−0.999 + 0.0202i)23-s − 0.200·25-s + 0.192i·27-s − 0.454·29-s + 0.553i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3359200672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3359200672\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.79 - 0.0971i)T \) |
| good | 7 | \( 1 + 4.84T + 7T^{2} \) |
| 11 | \( 1 + 3.09T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 + 1.04iT - 17T^{2} \) |
| 19 | \( 1 + 3.05T + 19T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 3.08iT - 31T^{2} \) |
| 37 | \( 1 + 3.78iT - 37T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 + 3.83iT - 47T^{2} \) |
| 53 | \( 1 + 8.14iT - 53T^{2} \) |
| 59 | \( 1 - 4.64iT - 59T^{2} \) |
| 61 | \( 1 - 11.3iT - 61T^{2} \) |
| 67 | \( 1 + 3.77T + 67T^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 2.53T + 73T^{2} \) |
| 79 | \( 1 + 1.04T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 + 3.89iT - 89T^{2} \) |
| 97 | \( 1 - 5.72iT - 97T^{2} \) |
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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
−8.059161675882536485470616067535, −7.34552813469010175772294755807, −6.82373614727070702727241329193, −6.01187099520731621902519535395, −5.45699033961699352037130263300, −4.51252272666447774523370465870, −3.55150235848560526592639487649, −2.71825895509536087166495051521, −2.05995542978126612898410931975, −0.42317292055213164144178485279,
0.19028270674210585963516531558, 2.28359755793792857030734355096, 2.82399788733433908982743417970, 3.62843166964713083813964786129, 4.37099290393586340582906358470, 5.32722936830464359138329705220, 6.08557298974710151765757432235, 6.59455563731878386568381886175, 7.45250601522713560415913406055, 8.054860489390846682295094605323
