L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 0.585·11-s + 16-s − 1.41·17-s − 2.82·19-s − 20-s + 0.585·22-s − 4.82·23-s + 25-s − 8.24·29-s + 5.07·31-s − 32-s + 1.41·34-s − 1.41·37-s + 2.82·38-s + 40-s + 8.82·41-s + 4.58·43-s − 0.585·44-s + 4.82·46-s + 9.07·47-s − 50-s + 9.31·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.176·11-s + 0.250·16-s − 0.342·17-s − 0.648·19-s − 0.223·20-s + 0.124·22-s − 1.00·23-s + 0.200·25-s − 1.53·29-s + 0.910·31-s − 0.176·32-s + 0.242·34-s − 0.232·37-s + 0.458·38-s + 0.158·40-s + 1.37·41-s + 0.699·43-s − 0.0883·44-s + 0.711·46-s + 1.32·47-s − 0.141·50-s + 1.27·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9088887485\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9088887485\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.585T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 4.58T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 7.89T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.351887475032721664037497919489, −7.66976992285672531886610698417, −7.15991630721081878821854163042, −6.20244475736694238574184177443, −5.64115580524938951581261519207, −4.45805013245717675182016087525, −3.83611390297570794866730714467, −2.71113509120715599062087712455, −1.90324895442588401192918624825, −0.57957080469630182225994034023,
0.57957080469630182225994034023, 1.90324895442588401192918624825, 2.71113509120715599062087712455, 3.83611390297570794866730714467, 4.45805013245717675182016087525, 5.64115580524938951581261519207, 6.20244475736694238574184177443, 7.15991630721081878821854163042, 7.66976992285672531886610698417, 8.351887475032721664037497919489