L(s) = 1 | − 3.25i·3-s + (0.386 + 2.20i)5-s + 1.44i·7-s − 7.62·9-s + 2.03·11-s + 0.557i·13-s + (7.17 − 1.25i)15-s + 3.91i·17-s − 6.73·19-s + 4.70·21-s − i·23-s + (−4.70 + 1.70i)25-s + 15.0i·27-s + 9.69·29-s + 3.07·31-s + ⋯ |
L(s) = 1 | − 1.88i·3-s + (0.172 + 0.984i)5-s + 0.545i·7-s − 2.54·9-s + 0.612·11-s + 0.154i·13-s + (1.85 − 0.325i)15-s + 0.950i·17-s − 1.54·19-s + 1.02·21-s − 0.208i·23-s + (−0.940 + 0.340i)25-s + 2.89i·27-s + 1.80·29-s + 0.552·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.434945494\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.434945494\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.386 - 2.20i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + 3.25iT - 3T^{2} \) |
| 7 | \( 1 - 1.44iT - 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 - 0.557iT - 13T^{2} \) |
| 17 | \( 1 - 3.91iT - 17T^{2} \) |
| 19 | \( 1 + 6.73T + 19T^{2} \) |
| 29 | \( 1 - 9.69T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 - 7.03T + 41T^{2} \) |
| 43 | \( 1 - 5.06iT - 43T^{2} \) |
| 47 | \( 1 - 0.659iT - 47T^{2} \) |
| 53 | \( 1 - 11.1iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.73T + 61T^{2} \) |
| 67 | \( 1 - 12.0iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 9.26iT - 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 5.25T + 89T^{2} \) |
| 97 | \( 1 - 0.0813iT - 97T^{2} \) |
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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
−8.884437011927465732523874682482, −8.385958010387751859616165763879, −7.61735613253926166618031651093, −6.67407615713014886595005666992, −6.38046045323592082135533685270, −5.77906458288743794623223085708, −4.22275390293608620202969110671, −2.83176528100669412459336396096, −2.29038338860313140629543203680, −1.22312984020402347322105115605,
0.57262405666992691812698506406, 2.46402197389089594655461471035, 3.70122764085551656922093456148, 4.34897666600668510329530432561, 4.87916280385220381448373060877, 5.72903369967559171427337947455, 6.66118764963117000560056841490, 8.034995833279724749218268511076, 8.790518100387555695827680769740, 9.172002978828924845509847491006