| L(s) = 1 | + 3-s + 5-s + 1.39·7-s + 9-s + 6.64·13-s + 15-s + 3.39·17-s + 1.39·21-s + 23-s + 25-s + 27-s + 0.601·29-s − 6.04·31-s + 1.39·35-s − 8.69·37-s + 6.64·39-s + 5.24·41-s − 7.44·43-s + 45-s + 2.79·47-s − 5.04·49-s + 3.39·51-s + 9.24·53-s − 3.24·59-s − 9.44·61-s + 1.39·63-s + 6.64·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.528·7-s + 0.333·9-s + 1.84·13-s + 0.258·15-s + 0.824·17-s + 0.305·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.111·29-s − 1.08·31-s + 0.236·35-s − 1.42·37-s + 1.06·39-s + 0.819·41-s − 1.13·43-s + 0.149·45-s + 0.407·47-s − 0.720·49-s + 0.475·51-s + 1.27·53-s − 0.422·59-s − 1.20·61-s + 0.176·63-s + 0.824·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.014017478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.014017478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.64T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 0.601T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 41 | \( 1 - 5.24T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 - 2.79T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + 3.24T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 - 4.79T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 6.09T + 89T^{2} \) |
| 97 | \( 1 - 9.44T + 97T^{2} \) |
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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
−8.774567539058877833958527592620, −8.194042898551758842504747480783, −7.41543501713353955659721091576, −6.51564650367522708173332042238, −5.73743678102741873679482828906, −4.97067616329237072165063894044, −3.83022006134738278016562742401, −3.25811507196515210993208745274, −1.98045531156101823126649434048, −1.17840817945433753252223274223,
1.17840817945433753252223274223, 1.98045531156101823126649434048, 3.25811507196515210993208745274, 3.83022006134738278016562742401, 4.97067616329237072165063894044, 5.73743678102741873679482828906, 6.51564650367522708173332042238, 7.41543501713353955659721091576, 8.194042898551758842504747480783, 8.774567539058877833958527592620
