| L(s) = 1 | + 3·3-s − 10.1·5-s + 9·9-s + 37.8·11-s + 74.0·13-s − 30.4·15-s + 44.6·17-s + 139.·19-s − 57.8·23-s − 21.8·25-s + 27·27-s + 39.5·29-s − 254.·31-s + 113.·33-s + 18.1·37-s + 222.·39-s + 420.·41-s + 382.·43-s − 91.4·45-s − 484.·47-s + 134.·51-s + 36.0·53-s − 383.·55-s + 418.·57-s − 145.·59-s − 78.1·61-s − 752.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.908·5-s + 0.333·9-s + 1.03·11-s + 1.58·13-s − 0.524·15-s + 0.637·17-s + 1.68·19-s − 0.524·23-s − 0.174·25-s + 0.192·27-s + 0.252·29-s − 1.47·31-s + 0.598·33-s + 0.0804·37-s + 0.912·39-s + 1.60·41-s + 1.35·43-s − 0.302·45-s − 1.50·47-s + 0.367·51-s + 0.0935·53-s − 0.941·55-s + 0.971·57-s − 0.321·59-s − 0.164·61-s − 1.43·65-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
| \(L(2)\) |
\(\approx\) |
\(3.047072141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.047072141\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 10.1T + 125T^{2} \) |
| 11 | \( 1 - 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 39.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 254.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 420.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 382.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 484.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 36.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 145.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 78.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 273.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 717.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 404.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 681.T + 9.12e5T^{2} \) |
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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
−8.678198847548087849921383730505, −7.73204691731937448992145081302, −7.43503885218563591551834849183, −6.30091673473803853746851965054, −5.59550012074656380189609105971, −4.30964654533802490617324426283, −3.69213941999899749650639429257, −3.13989103882122665967192624455, −1.63249503149926929044577535102, −0.818015891927076990246539509357,
0.818015891927076990246539509357, 1.63249503149926929044577535102, 3.13989103882122665967192624455, 3.69213941999899749650639429257, 4.30964654533802490617324426283, 5.59550012074656380189609105971, 6.30091673473803853746851965054, 7.43503885218563591551834849183, 7.73204691731937448992145081302, 8.678198847548087849921383730505
