| L(s) = 1 | − 5-s + 7-s + 11-s − 6·13-s − 3·17-s − 5·19-s + 2·23-s + 25-s + 5·29-s + 5·31-s − 35-s − 37-s + 2·41-s + 12·43-s + 2·47-s − 6·49-s + 13·53-s − 55-s − 2·59-s + 61-s + 6·65-s + 16·67-s − 15·71-s + 10·73-s + 77-s + 2·79-s + 14·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.301·11-s − 1.66·13-s − 0.727·17-s − 1.14·19-s + 0.417·23-s + 1/5·25-s + 0.928·29-s + 0.898·31-s − 0.169·35-s − 0.164·37-s + 0.312·41-s + 1.82·43-s + 0.291·47-s − 6/7·49-s + 1.78·53-s − 0.134·55-s − 0.260·59-s + 0.128·61-s + 0.744·65-s + 1.95·67-s − 1.78·71-s + 1.17·73-s + 0.113·77-s + 0.225·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440004295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440004295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.438674817423357570936470035554, −7.73433059133812301192977364984, −6.99970383505905243647391751086, −6.42321132891736916212149036139, −5.34122428081478900562427414965, −4.56735255286468441470482194742, −4.09982993045677391676142437218, −2.78789847884678517090591066257, −2.14467665127329154435692730090, −0.67084152583760764906013623104,
0.67084152583760764906013623104, 2.14467665127329154435692730090, 2.78789847884678517090591066257, 4.09982993045677391676142437218, 4.56735255286468441470482194742, 5.34122428081478900562427414965, 6.42321132891736916212149036139, 6.99970383505905243647391751086, 7.73433059133812301192977364984, 8.438674817423357570936470035554
