L(s) = 1 | − 0.618·2-s − 3-s − 1.61·4-s + 0.618·6-s + 2.23·8-s + 9-s + 2.23·11-s + 1.61·12-s − 2.47·13-s + 1.85·16-s + 6.47·17-s − 0.618·18-s + 6.47·19-s − 1.38·22-s − 0.236·23-s − 2.23·24-s + 1.52·26-s − 27-s − 3·29-s + 4·31-s − 5.61·32-s − 2.23·33-s − 4.00·34-s − 1.61·36-s + 5.47·37-s − 4.00·38-s + 2.47·39-s + ⋯ |
L(s) = 1 | − 0.437·2-s − 0.577·3-s − 0.809·4-s + 0.252·6-s + 0.790·8-s + 0.333·9-s + 0.674·11-s + 0.467·12-s − 0.685·13-s + 0.463·16-s + 1.56·17-s − 0.145·18-s + 1.48·19-s − 0.294·22-s − 0.0492·23-s − 0.456·24-s + 0.299·26-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.993·32-s − 0.389·33-s − 0.685·34-s − 0.269·36-s + 0.899·37-s − 0.648·38-s + 0.395·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.025346468\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.025346468\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 0.236T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 0.708T + 67T^{2} \) |
| 71 | \( 1 - 3.76T + 71T^{2} \) |
| 73 | \( 1 + 5.52T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.501951421340583370945141673758, −7.80032570571314572026789764339, −7.23713402589484232943368699180, −6.29083246312614561869923438872, −5.30852168681852901648024062341, −4.98624811114539665539354289203, −3.91037351924812594223985792086, −3.16537886852318153350254269490, −1.57050684715428305011054075461, −0.71075795450917434645347341932,
0.71075795450917434645347341932, 1.57050684715428305011054075461, 3.16537886852318153350254269490, 3.91037351924812594223985792086, 4.98624811114539665539354289203, 5.30852168681852901648024062341, 6.29083246312614561869923438872, 7.23713402589484232943368699180, 7.80032570571314572026789764339, 8.501951421340583370945141673758