| L(s) = 1 | − 5.36·2-s + 20.7·4-s + 2.69·5-s − 15.2·7-s − 68.5·8-s − 14.4·10-s − 66.8·11-s + 81.5·14-s + 201.·16-s − 4.16·17-s − 26.0·19-s + 56.0·20-s + 358.·22-s − 47.3·23-s − 117.·25-s − 315.·28-s − 257.·29-s − 206.·31-s − 532.·32-s + 22.3·34-s − 40.9·35-s − 175.·37-s + 139.·38-s − 184.·40-s + 156.·41-s + 51.9·43-s − 1.38e3·44-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.59·4-s + 0.241·5-s − 0.820·7-s − 3.03·8-s − 0.457·10-s − 1.83·11-s + 1.55·14-s + 3.14·16-s − 0.0594·17-s − 0.314·19-s + 0.626·20-s + 3.47·22-s − 0.429·23-s − 0.941·25-s − 2.13·28-s − 1.64·29-s − 1.19·31-s − 2.94·32-s + 0.112·34-s − 0.197·35-s − 0.780·37-s + 0.597·38-s − 0.730·40-s + 0.595·41-s + 0.184·43-s − 4.76·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09282858832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09282858832\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.36T + 8T^{2} \) |
| 5 | \( 1 - 2.69T + 125T^{2} \) |
| 7 | \( 1 + 15.2T + 343T^{2} \) |
| 11 | \( 1 + 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 4.16T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 47.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 51.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 10.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 119.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 22.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 740.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 215.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−9.235695637149367062276442396984, −8.325191510381031168749681038569, −7.67835121382867349463344855767, −7.07083835306118543686341353343, −6.07218609074005622978774543587, −5.40148964468771628898345430029, −3.55865268352195665765302709292, −2.52458619512019929331950559126, −1.79271479306014364518367995735, −0.18480224497698183056356409812,
0.18480224497698183056356409812, 1.79271479306014364518367995735, 2.52458619512019929331950559126, 3.55865268352195665765302709292, 5.40148964468771628898345430029, 6.07218609074005622978774543587, 7.07083835306118543686341353343, 7.67835121382867349463344855767, 8.325191510381031168749681038569, 9.235695637149367062276442396984
