L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 2·11-s + 12-s + 13-s + 16-s + 18-s − 19-s + 2·22-s + 8·23-s + 24-s + 26-s + 27-s + 10·29-s + 8·31-s + 32-s + 2·33-s + 36-s − 4·37-s − 38-s + 39-s + 5·41-s − 8·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.426·22-s + 1.66·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.176·32-s + 0.348·33-s + 1/6·36-s − 0.657·37-s − 0.162·38-s + 0.160·39-s + 0.780·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.784957234\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.784957234\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.60862563987772, −12.96708748048067, −12.47215674345962, −12.13365029999538, −11.60552308596633, −10.99912033155541, −10.66880987638394, −10.01666563974852, −9.608282262131815, −8.957513810208728, −8.472162876669553, −8.189616873778196, −7.393888990414382, −6.902847025053921, −6.536599503157864, −6.036513004339087, −5.317391955565131, −4.620126375196535, −4.503011951878676, −3.678221374619406, −3.122422157430666, −2.764431572496421, −2.045694291499211, −1.252847651379002, −0.7716031325923583,
0.7716031325923583, 1.252847651379002, 2.045694291499211, 2.764431572496421, 3.122422157430666, 3.678221374619406, 4.503011951878676, 4.620126375196535, 5.317391955565131, 6.036513004339087, 6.536599503157864, 6.902847025053921, 7.393888990414382, 8.189616873778196, 8.472162876669553, 8.957513810208728, 9.608282262131815, 10.01666563974852, 10.66880987638394, 10.99912033155541, 11.60552308596633, 12.13365029999538, 12.47215674345962, 12.96708748048067, 13.60862563987772