L(s) = 1 | + (−0.276 − 0.479i)5-s + (−1.98 + 1.74i)7-s + (−4.03 − 2.32i)11-s + (3.58 − 2.06i)13-s + 7.24·17-s − 6.71i·19-s + (4.85 − 2.80i)23-s + (2.34 − 4.06i)25-s + (−1.16 − 0.673i)29-s + (−0.830 + 0.479i)31-s + (1.38 + 0.469i)35-s − 7.06·37-s + (2.39 + 4.14i)41-s + (−1.02 + 1.78i)43-s + (4.90 − 8.49i)47-s + ⋯ |
L(s) = 1 | + (−0.123 − 0.214i)5-s + (−0.751 + 0.660i)7-s + (−1.21 − 0.702i)11-s + (0.993 − 0.573i)13-s + 1.75·17-s − 1.54i·19-s + (1.01 − 0.584i)23-s + (0.469 − 0.812i)25-s + (−0.216 − 0.125i)29-s + (−0.149 + 0.0861i)31-s + (0.234 + 0.0793i)35-s − 1.16·37-s + (0.373 + 0.646i)41-s + (−0.156 + 0.271i)43-s + (0.715 − 1.23i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04760 - 0.644664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04760 - 0.644664i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.98 - 1.74i)T \) |
good | 5 | \( 1 + (0.276 + 0.479i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (4.03 + 2.32i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.58 + 2.06i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 7.24T + 17T^{2} \) |
| 19 | \( 1 + 6.71iT - 19T^{2} \) |
| 23 | \( 1 + (-4.85 + 2.80i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.16 + 0.673i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.830 - 0.479i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + (-2.39 - 4.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.02 - 1.78i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.90 + 8.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.43iT - 53T^{2} \) |
| 59 | \( 1 + (3.89 + 6.75i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.37 + 3.10i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.68 - 2.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.407iT - 71T^{2} \) |
| 73 | \( 1 + 8.63iT - 73T^{2} \) |
| 79 | \( 1 + (0.318 - 0.551i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.78 - 4.82i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6.93T + 89T^{2} \) |
| 97 | \( 1 + (-7.48 - 4.32i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.34686048581263278900206646471, −9.205012432115912858305335312557, −8.544814344255380722965026067462, −7.74001679926107113366834639516, −6.59634205783695648529924389708, −5.66527087995487092293502427269, −4.98976354834236041798883160067, −3.36670701175616138789626599968, −2.74736569120403460254186798997, −0.68514655919505801476605530315,
1.42084481034554037945961702400, 3.14570656567761913140787819128, 3.82327559243249823341419613146, 5.21790361999317593710318935469, 6.03742952607755706614062312830, 7.26663658655598962336173229724, 7.63642182949181495229462739076, 8.833448316709017950338432726533, 9.852858798732793060619880382669, 10.38320764744152736683638237637