| L(s) = 1 | + 0.585·5-s − 3.41·7-s − 2·11-s − 2.82·13-s − 3.65·17-s − 5.65·19-s + 1.17·23-s − 4.65·25-s + 0.585·29-s − 4.58·31-s − 2·35-s + 9.65·37-s + 11.6·41-s + 1.65·43-s + 12.4·47-s + 4.65·49-s + 11.8·53-s − 1.17·55-s + 4·59-s + 9.65·61-s − 1.65·65-s − 8·67-s + 9.17·71-s − 1.65·73-s + 6.82·77-s − 5.75·79-s − 9.31·83-s + ⋯ |
| L(s) = 1 | + 0.261·5-s − 1.29·7-s − 0.603·11-s − 0.784·13-s − 0.886·17-s − 1.29·19-s + 0.244·23-s − 0.931·25-s + 0.108·29-s − 0.823·31-s − 0.338·35-s + 1.58·37-s + 1.82·41-s + 0.252·43-s + 1.82·47-s + 0.665·49-s + 1.63·53-s − 0.157·55-s + 0.520·59-s + 1.23·61-s − 0.205·65-s − 0.977·67-s + 1.08·71-s − 0.193·73-s + 0.778·77-s − 0.647·79-s − 1.02·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007924952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007924952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 + 1.65T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.375329618262859545512598849158, −7.43908389151436128273852012116, −6.92002365114007219002765867165, −6.04242688218214418044926943561, −5.62611463452801798363922552433, −4.43301874502472002572152263115, −3.89011414365806996401515786361, −2.61019775133102439087139649584, −2.30406117315868249184316057808, −0.52078042396469810305240631539,
0.52078042396469810305240631539, 2.30406117315868249184316057808, 2.61019775133102439087139649584, 3.89011414365806996401515786361, 4.43301874502472002572152263115, 5.62611463452801798363922552433, 6.04242688218214418044926943561, 6.92002365114007219002765867165, 7.43908389151436128273852012116, 8.375329618262859545512598849158
