| L(s) = 1 | + i·2-s + 3-s − 4-s − 2.26i·5-s + i·6-s − i·8-s + 9-s + 2.26·10-s + 1.26i·11-s − 12-s + (−1.26 + 3.37i)13-s − 2.26i·15-s + 16-s + 0.377·17-s + i·18-s − 7.75i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 1.01i·5-s + 0.408i·6-s − 0.353i·8-s + 0.333·9-s + 0.715·10-s + 0.380i·11-s − 0.288·12-s + (−0.349 + 0.936i)13-s − 0.583i·15-s + 0.250·16-s + 0.0915·17-s + 0.235i·18-s − 1.77i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.061064728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061064728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (1.26 - 3.37i)T \) |
| good | 5 | \( 1 + 2.26iT - 5T^{2} \) |
| 11 | \( 1 - 1.26iT - 11T^{2} \) |
| 17 | \( 1 - 0.377T + 17T^{2} \) |
| 19 | \( 1 + 7.75iT - 19T^{2} \) |
| 23 | \( 1 - 4.49T + 23T^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 3.37iT - 37T^{2} \) |
| 41 | \( 1 + 9.90iT - 41T^{2} \) |
| 43 | \( 1 - 8.75T + 43T^{2} \) |
| 47 | \( 1 - 3.29iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 4.66iT - 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 - 1.29iT - 67T^{2} \) |
| 71 | \( 1 - 4.11iT - 71T^{2} \) |
| 73 | \( 1 + 9.54iT - 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 + 6.39iT - 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 8.03iT - 97T^{2} \) |
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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
−8.767668678578858871754935721887, −7.48031696555652127143518798648, −7.24538125370882803953165628102, −6.36446210184708581346921807832, −5.27089329707637062440312392943, −4.73376244657454827494585023308, −4.13002818986707835171419699312, −2.95323755104187274266227755803, −1.87044353664246547875098029223, −0.63736043820575761040223639318,
1.09309159503768249338593532093, 2.28308034938824237941834987359, 3.06158194969221604971613049908, 3.53573228931598763207853081305, 4.51331050064834871360007311724, 5.58146931772949346390551895581, 6.24480375614693852334854136991, 7.30615157569567011529621318113, 7.84268563736879827889252654360, 8.526987094865977716156709761424
