| L(s) = 1 | + 1.30·2-s − 0.302·4-s − 5-s + 7-s − 3·8-s − 1.30·10-s − 5.60·11-s + 1.30·14-s − 3.30·16-s − 0.394·17-s − 1.60·19-s + 0.302·20-s − 7.30·22-s + 3·23-s + 25-s − 0.302·28-s − 8.21·29-s + 4·31-s + 1.69·32-s − 0.513·34-s − 35-s + 3.60·37-s − 2.09·38-s + 3·40-s + 3·41-s + 4.21·43-s + 1.69·44-s + ⋯ |
| L(s) = 1 | + 0.921·2-s − 0.151·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.411·10-s − 1.69·11-s + 0.348·14-s − 0.825·16-s − 0.0956·17-s − 0.368·19-s + 0.0677·20-s − 1.55·22-s + 0.625·23-s + 0.200·25-s − 0.0572·28-s − 1.52·29-s + 0.718·31-s + 0.300·32-s − 0.0881·34-s − 0.169·35-s + 0.592·37-s − 0.339·38-s + 0.474·40-s + 0.468·41-s + 0.642·43-s + 0.255·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.558798362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.558798362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 17 | \( 1 + 0.394T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 16.8T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 - 8.21T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.900935424269863812101493754382, −7.21590865983254336304879529180, −6.25539395746621548437920960988, −5.60492692809824490267219292098, −4.90774556585015164398633868986, −4.52478729076409106861685138615, −3.57864642965645565725350175716, −2.91930574685510432394674960332, −2.08874436600015990239056791943, −0.51798782905921985681394034612,
0.51798782905921985681394034612, 2.08874436600015990239056791943, 2.91930574685510432394674960332, 3.57864642965645565725350175716, 4.52478729076409106861685138615, 4.90774556585015164398633868986, 5.60492692809824490267219292098, 6.25539395746621548437920960988, 7.21590865983254336304879529180, 7.900935424269863812101493754382
