L(s) = 1 | + 2-s + (0.403 − 1.68i)3-s + 4-s + (0.403 − 1.68i)6-s + (−1.28 − 2.31i)7-s + 8-s + (−2.67 − 1.35i)9-s − 5.34i·11-s + (0.403 − 1.68i)12-s − 3.95·13-s + (−1.28 − 2.31i)14-s + 16-s + 7.32i·17-s + (−2.67 − 1.35i)18-s − 0.807i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.232 − 0.972i)3-s + 0.5·4-s + (0.164 − 0.687i)6-s + (−0.484 − 0.874i)7-s + 0.353·8-s + (−0.891 − 0.453i)9-s − 1.61i·11-s + (0.116 − 0.486i)12-s − 1.09·13-s + (−0.342 − 0.618i)14-s + 0.250·16-s + 1.77i·17-s + (−0.630 − 0.320i)18-s − 0.185i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.002968182\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.002968182\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.403 + 1.68i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.28 + 2.31i)T \) |
good | 11 | \( 1 + 5.34iT - 11T^{2} \) |
| 13 | \( 1 + 3.95T + 13T^{2} \) |
| 17 | \( 1 - 7.32iT - 17T^{2} \) |
| 19 | \( 1 + 0.807iT - 19T^{2} \) |
| 23 | \( 1 - 0.281T + 23T^{2} \) |
| 29 | \( 1 + 0.281iT - 29T^{2} \) |
| 31 | \( 1 + 9.07iT - 31T^{2} \) |
| 37 | \( 1 - 6.06iT - 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 + 6.34iT - 43T^{2} \) |
| 47 | \( 1 + 5.78iT - 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 6.71iT - 67T^{2} \) |
| 71 | \( 1 - 3.36iT - 71T^{2} \) |
| 73 | \( 1 - 4.98T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 - 1.53iT - 83T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.644005146072060601862004850378, −8.459101737264051080079858547916, −7.85687838292642386323011606208, −6.93374482605554223846617135591, −6.22393416973335925376176730554, −5.52162348865147596999621572300, −4.05093650676114646160321016169, −3.29774769657889938243974637335, −2.17574730889154037426438247923, −0.65390603369604191812260471002,
2.35466556014619289450650766168, 2.91535021999006765735239057645, 4.25709061785693564008981012694, 4.96911641944236216142275621522, 5.57488301386996439597583898953, 6.90317486988857774282375792620, 7.52721139489974665202822095166, 8.851182606931875991402763021994, 9.590532049094183900377068667037, 10.00364130813080729233769512417