| L(s) = 1 | + 3·3-s + 7.63·5-s + 9·9-s + 30.8·11-s + 42.5·13-s + 22.8·15-s − 4.36·17-s + 27.2·19-s + 56.6·23-s − 66.7·25-s + 27·27-s + 219.·29-s + 172.·31-s + 92.6·33-s − 31.3·37-s + 127.·39-s − 274.·41-s − 188·43-s + 68.6·45-s + 50.0·47-s − 13.1·51-s + 282.·53-s + 235.·55-s + 81.7·57-s − 64.1·59-s + 97.1·61-s + 324.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.682·5-s + 0.333·9-s + 0.846·11-s + 0.907·13-s + 0.394·15-s − 0.0623·17-s + 0.329·19-s + 0.513·23-s − 0.534·25-s + 0.192·27-s + 1.40·29-s + 0.998·31-s + 0.488·33-s − 0.139·37-s + 0.523·39-s − 1.04·41-s − 0.666·43-s + 0.227·45-s + 0.155·47-s − 0.0359·51-s + 0.733·53-s + 0.577·55-s + 0.190·57-s − 0.141·59-s + 0.203·61-s + 0.619·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.071973270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.071973270\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 7.63T + 125T^{2} \) |
| 11 | \( 1 - 30.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.36T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 31.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 188T + 7.95e4T^{2} \) |
| 47 | \( 1 - 50.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 282.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 64.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 97.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 716.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 641.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 391.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 952.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 360.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 58.5T + 9.12e5T^{2} \) |
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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
−8.616264911319696441749326657060, −8.103928157901428692543209025358, −6.92450852338277697168669797679, −6.44131760222373719562418422969, −5.55283029209864577739781035708, −4.56694956050276462756508106554, −3.67513050921437170518731192167, −2.82921431103529530100965385810, −1.75788077709591370422214576444, −0.936793170367792303702327907250,
0.936793170367792303702327907250, 1.75788077709591370422214576444, 2.82921431103529530100965385810, 3.67513050921437170518731192167, 4.56694956050276462756508106554, 5.55283029209864577739781035708, 6.44131760222373719562418422969, 6.92450852338277697168669797679, 8.103928157901428692543209025358, 8.616264911319696441749326657060
