| L(s) = 1 | + 7.44·3-s − 18.2·5-s + 6.49·7-s + 28.4·9-s + 43.1·11-s + 9.88·13-s − 136.·15-s − 31.3·17-s + 19·19-s + 48.3·21-s − 29.5·23-s + 209.·25-s + 10.6·27-s − 29.5·29-s − 30.0·31-s + 321.·33-s − 118.·35-s + 151.·37-s + 73.5·39-s + 477.·41-s + 127.·43-s − 519.·45-s − 156.·47-s − 300.·49-s − 233.·51-s + 397.·53-s − 788.·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s − 1.63·5-s + 0.350·7-s + 1.05·9-s + 1.18·11-s + 0.210·13-s − 2.34·15-s − 0.447·17-s + 0.229·19-s + 0.502·21-s − 0.267·23-s + 1.67·25-s + 0.0759·27-s − 0.189·29-s − 0.173·31-s + 1.69·33-s − 0.573·35-s + 0.672·37-s + 0.302·39-s + 1.81·41-s + 0.453·43-s − 1.72·45-s − 0.484·47-s − 0.877·49-s − 0.641·51-s + 1.03·53-s − 1.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.929143956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929143956\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 7.44T + 27T^{2} \) |
| 5 | \( 1 + 18.2T + 125T^{2} \) |
| 7 | \( 1 - 6.49T + 343T^{2} \) |
| 11 | \( 1 - 43.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.88T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 29.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 29.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 30.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 477.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 397.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 271.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 709.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 765.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 239.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 578.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 85.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 739.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 714.T + 9.12e5T^{2} \) |
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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
−8.953468222655041654680616743907, −8.647827433463589919215962276494, −7.70001663618757038033129741997, −7.37545821256052242724503423382, −6.21467943835631525361039172975, −4.61699426246300252871787412458, −3.92424219259267608474237510734, −3.36067532057901490638191831686, −2.18429913712771753206883656607, −0.831519056349182815956726965792,
0.831519056349182815956726965792, 2.18429913712771753206883656607, 3.36067532057901490638191831686, 3.92424219259267608474237510734, 4.61699426246300252871787412458, 6.21467943835631525361039172975, 7.37545821256052242724503423382, 7.70001663618757038033129741997, 8.647827433463589919215962276494, 8.953468222655041654680616743907
