| 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
| \(L(1)\) |
\(\approx\) |
\(2.650143274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.650143274\) |
| \(L(\frac{3}{2})\) |
|
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
−8.160080707283992202933923098848, −7.53419064274351981000497546627, −6.92840228900874157262142488996, −5.62727862731267290650307035817, −5.21076955086117476672698831272, −4.50929241208421410222531756368, −4.00271766175966704242198957363, −2.75597185534434651999821651948, −1.82820872055293867223018615502, −0.855251217181061686728592280293,
1.01168468848430923748269190015, 1.80363997303794856672112572002, 2.61585734854458760936061117551, 3.66356730454853034176289733873, 4.70876991475755596324253585988, 5.12002949659959517528988368617, 6.04836516112801190580600194474, 6.93349899488476770314085622779, 7.48589197377165484529494259812, 7.923724183995082551897631741188
