| L(s) = 1 | + 2-s + 0.652·3-s + 4-s + 1.22·5-s + 0.652·6-s − 3.06·7-s + 8-s − 2.57·9-s + 1.22·10-s + 4.63·11-s + 0.652·12-s − 5.94·13-s − 3.06·14-s + 0.800·15-s + 16-s − 0.162·17-s − 2.57·18-s + 8.12·19-s + 1.22·20-s − 1.99·21-s + 4.63·22-s − 8.17·23-s + 0.652·24-s − 3.49·25-s − 5.94·26-s − 3.63·27-s − 3.06·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.376·3-s + 0.5·4-s + 0.548·5-s + 0.266·6-s − 1.15·7-s + 0.353·8-s − 0.857·9-s + 0.387·10-s + 1.39·11-s + 0.188·12-s − 1.64·13-s − 0.818·14-s + 0.206·15-s + 0.250·16-s − 0.0394·17-s − 0.606·18-s + 1.86·19-s + 0.274·20-s − 0.436·21-s + 0.988·22-s − 1.70·23-s + 0.133·24-s − 0.699·25-s − 1.16·26-s − 0.700·27-s − 0.579·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.665064264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665064264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 - 0.652T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 + 3.06T + 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 + 5.94T + 13T^{2} \) |
| 17 | \( 1 + 0.162T + 17T^{2} \) |
| 19 | \( 1 - 8.12T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 - 9.88T + 37T^{2} \) |
| 41 | \( 1 + 3.06T + 41T^{2} \) |
| 43 | \( 1 - 0.857T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 - 0.411T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 - 8.41T + 61T^{2} \) |
| 71 | \( 1 - 5.92T + 71T^{2} \) |
| 73 | \( 1 + 0.554T + 73T^{2} \) |
| 79 | \( 1 - 2.45T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 7.84T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−13.50262276029584750873091899968, −12.17461292352112062310054580766, −11.67173788509607065111980138819, −9.765182193751243799527511543014, −9.506635567291799298891089869849, −7.72347245354162789776578563768, −6.48070850409950107838790231667, −5.53443502916545735270144003276, −3.80616459225614149579450286321, −2.53099351775365002507038256065,
2.53099351775365002507038256065, 3.80616459225614149579450286321, 5.53443502916545735270144003276, 6.48070850409950107838790231667, 7.72347245354162789776578563768, 9.506635567291799298891089869849, 9.765182193751243799527511543014, 11.67173788509607065111980138819, 12.17461292352112062310054580766, 13.50262276029584750873091899968
