L(s) = 1 | + (−4.19 + 2.72i)5-s + 3.43i·7-s − 12.8i·11-s + 1.79i·13-s + 30.9·17-s − 19.2·19-s − 2.38·23-s + (10.1 − 22.8i)25-s + 35.8i·29-s − 40.9·31-s + (−9.36 − 14.4i)35-s − 53.6i·37-s + 2.48i·41-s − 55.1i·43-s + 57.0·47-s + ⋯ |
L(s) = 1 | + (−0.838 + 0.544i)5-s + 0.491i·7-s − 1.16i·11-s + 0.137i·13-s + 1.82·17-s − 1.01·19-s − 0.103·23-s + (0.406 − 0.913i)25-s + 1.23i·29-s − 1.31·31-s + (−0.267 − 0.411i)35-s − 1.44i·37-s + 0.0605i·41-s − 1.28i·43-s + 1.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.512987989\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.512987989\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.19 - 2.72i)T \) |
good | 7 | \( 1 - 3.43iT - 49T^{2} \) |
| 11 | \( 1 + 12.8iT - 121T^{2} \) |
| 13 | \( 1 - 1.79iT - 169T^{2} \) |
| 17 | \( 1 - 30.9T + 289T^{2} \) |
| 19 | \( 1 + 19.2T + 361T^{2} \) |
| 23 | \( 1 + 2.38T + 529T^{2} \) |
| 29 | \( 1 - 35.8iT - 841T^{2} \) |
| 31 | \( 1 + 40.9T + 961T^{2} \) |
| 37 | \( 1 + 53.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 2.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 57.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 19.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 69.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 17.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 51.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 53.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 42.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 88.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 28.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 68.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 156. iT - 9.40e3T^{2} \) |
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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
−8.998536304114720384375696940320, −8.590760726626657903778790682094, −7.62365190339009202301319820165, −7.04865331634098276029272507168, −5.89310167068354414722124943686, −5.39651873694828985857044212733, −3.94897523892735763023669522920, −3.41982423723452530710782706669, −2.34580326249118914339665430271, −0.76280800634013802855425276622,
0.62349348681045542971580214731, 1.83749527698529773290310877034, 3.28118953595573726027328265602, 4.14891937137631979571978573125, 4.82712496905464439100896975225, 5.81596217085898441396549653682, 6.91036451559497564330605026722, 7.71933695129022405772944572055, 8.078467713167268468313643518240, 9.188596513241518448775098959712