L(s) = 1 | − 2·7-s + 2·11-s − 13-s − 2·17-s − 2·19-s − 8·23-s − 6·29-s − 2·31-s + 2·37-s + 2·41-s + 6·47-s − 3·49-s + 10·53-s − 14·59-s + 10·61-s + 2·67-s + 6·71-s − 2·73-s − 4·77-s − 12·79-s − 6·83-s + 18·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s − 1.66·23-s − 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.312·41-s + 0.875·47-s − 3/7·49-s + 1.37·53-s − 1.82·59-s + 1.28·61-s + 0.244·67-s + 0.712·71-s − 0.234·73-s − 0.455·77-s − 1.35·79-s − 0.658·83-s + 1.90·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7150371390\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7150371390\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.08995151600556, −12.70316827924520, −12.17036774609701, −11.77287485027514, −11.31070075247012, −10.71485576093195, −10.27481310834470, −9.798820985237851, −9.303569218327328, −8.978417660458138, −8.405415052835719, −7.735149989493755, −7.457207034603121, −6.601112445195715, −6.532311793248655, −5.762372583612470, −5.503575288060522, −4.616996147469716, −4.085651675377716, −3.776717660524260, −3.117660672134683, −2.330914186303811, −2.005512526458107, −1.163002162075334, −0.2464236385882396,
0.2464236385882396, 1.163002162075334, 2.005512526458107, 2.330914186303811, 3.117660672134683, 3.776717660524260, 4.085651675377716, 4.616996147469716, 5.503575288060522, 5.762372583612470, 6.532311793248655, 6.601112445195715, 7.457207034603121, 7.735149989493755, 8.405415052835719, 8.978417660458138, 9.303569218327328, 9.798820985237851, 10.27481310834470, 10.71485576093195, 11.31070075247012, 11.77287485027514, 12.17036774609701, 12.70316827924520, 13.08995151600556