| L(s) = 1 | + (0.692 + 1.23i)2-s + (−1.04 + 1.70i)4-s − i·5-s − i·7-s + (−2.82 − 0.0992i)8-s + (1.23 − 0.692i)10-s − 4.57·11-s − 0.572·13-s + (1.23 − 0.692i)14-s + (−1.83 − 3.55i)16-s − 4.58i·17-s + 0.824i·19-s + (1.70 + 1.04i)20-s + (−3.17 − 5.64i)22-s + 2.54·23-s + ⋯ |
| L(s) = 1 | + (0.489 + 0.871i)2-s + (−0.520 + 0.854i)4-s − 0.447i·5-s − 0.377i·7-s + (−0.999 − 0.0350i)8-s + (0.389 − 0.219i)10-s − 1.37·11-s − 0.158·13-s + (0.329 − 0.185i)14-s + (−0.458 − 0.888i)16-s − 1.11i·17-s + 0.189i·19-s + (0.381 + 0.232i)20-s + (−0.675 − 1.20i)22-s + 0.531·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8670261798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8670261798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.692 - 1.23i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 + 0.572T + 13T^{2} \) |
| 17 | \( 1 + 4.58iT - 17T^{2} \) |
| 19 | \( 1 - 0.824iT - 19T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 + 9.42iT - 29T^{2} \) |
| 31 | \( 1 + 2.63iT - 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 - 5.36iT - 41T^{2} \) |
| 43 | \( 1 + 3.04iT - 43T^{2} \) |
| 47 | \( 1 + 7.71T + 47T^{2} \) |
| 53 | \( 1 + 13.6iT - 53T^{2} \) |
| 59 | \( 1 + 15.3T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 1.72iT - 67T^{2} \) |
| 71 | \( 1 + 8.57T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 79 | \( 1 - 7.09iT - 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 + 9.14iT - 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−9.482406008098779487250169058151, −8.421254149209474006469772703646, −7.83176671842130365830015837583, −7.13988787202143024325681983784, −6.16099093008871751993151909525, −5.17463068032003991594725877979, −4.72485395523076161509067913014, −3.54653741239538348866580522326, −2.46682281001879873357449747653, −0.29533089861959580288737061145,
1.65403703702836911582957409559, 2.76305222669579669845635159400, 3.46792517906499834972836621105, 4.75785170062732036829567458864, 5.41422091251826976738724824817, 6.31497867378558883444582754147, 7.35026592809445302438121412068, 8.431160452640625035483724717155, 9.123021088360605421969020391692, 10.19331653123672918167380450826
