| L(s) = 1 | + (−0.987 + 0.156i)2-s + (−1.00 + 1.97i)3-s + (0.951 − 0.309i)4-s + (1.74 − 1.40i)5-s + (0.684 − 2.10i)6-s + (−1.24 − 1.24i)7-s + (−0.891 + 0.453i)8-s + (−1.12 − 1.54i)9-s + (−1.50 + 1.65i)10-s + (−0.307 − 0.223i)11-s + (−0.346 + 2.18i)12-s + (−0.429 − 0.0679i)13-s + (1.42 + 1.03i)14-s + (1.01 + 4.84i)15-s + (0.809 − 0.587i)16-s + (1.50 − 0.769i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 + 0.110i)2-s + (−0.580 + 1.13i)3-s + (0.475 − 0.154i)4-s + (0.779 − 0.626i)5-s + (0.279 − 0.859i)6-s + (−0.470 − 0.470i)7-s + (−0.315 + 0.160i)8-s + (−0.373 − 0.513i)9-s + (−0.474 + 0.523i)10-s + (−0.0927 − 0.0673i)11-s + (−0.100 + 0.631i)12-s + (−0.119 − 0.0188i)13-s + (0.380 + 0.276i)14-s + (0.261 + 1.25i)15-s + (0.202 − 0.146i)16-s + (0.366 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.570476 - 0.334799i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.570476 - 0.334799i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 - 0.156i)T \) |
| 5 | \( 1 + (-1.74 + 1.40i)T \) |
| 19 | \( 1 + (0.284 + 4.34i)T \) |
| good | 3 | \( 1 + (1.00 - 1.97i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (1.24 + 1.24i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.307 + 0.223i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.429 + 0.0679i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.50 + 0.769i)T + (9.99 - 13.7i)T^{2} \) |
| 23 | \( 1 + (4.31 - 0.683i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (2.32 + 7.14i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.00 + 0.975i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.0275 + 0.174i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (0.974 + 1.34i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.98 - 2.98i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.17 + 4.27i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.410 - 0.806i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.49 + 3.98i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.15 - 3.01i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.95 + 5.80i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-8.79 + 2.85i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.46 + 0.231i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (1.48 + 4.57i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.78 + 3.50i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.691 - 0.502i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.55 + 1.30i)T + (57.0 + 78.4i)T^{2} \) |
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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
−9.817449766445539753807447377393, −9.434785182795152934494213789985, −8.472450474822431976057930041971, −7.42174983704718416497697564694, −6.33355858475284514945145212720, −5.55363023938154729207928801797, −4.73766872928332202704280003533, −3.69143705088834778028863333356, −2.15456330688453823713236821535, −0.41842917595918664840842910315,
1.42274163004042878693342483684, 2.30555787199166076471490254368, 3.52631079268558969734113245536, 5.50702929949895163747313922718, 6.06637734875002666031122273667, 6.83548026717389274503050416336, 7.49099234033101577960283374249, 8.462322233019006616341645351173, 9.488619046604127774933721813910, 10.13212906786603071767814236908
