| L(s) = 1 | − 2.73·2-s + 3-s + 5.50·4-s + 3.38·5-s − 2.73·6-s − 9.61·8-s + 9-s − 9.26·10-s + 1.88·11-s + 5.50·12-s − 1.76·13-s + 3.38·15-s + 15.3·16-s + 5.89·17-s − 2.73·18-s + 6.62·19-s + 18.6·20-s − 5.16·22-s + 23-s − 9.61·24-s + 6.43·25-s + 4.84·26-s + 27-s + 8.24·29-s − 9.26·30-s − 2.94·31-s − 22.7·32-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.577·3-s + 2.75·4-s + 1.51·5-s − 1.11·6-s − 3.39·8-s + 0.333·9-s − 2.93·10-s + 0.567·11-s + 1.58·12-s − 0.490·13-s + 0.873·15-s + 3.82·16-s + 1.42·17-s − 0.645·18-s + 1.52·19-s + 4.16·20-s − 1.10·22-s + 0.208·23-s − 1.96·24-s + 1.28·25-s + 0.950·26-s + 0.192·27-s + 1.53·29-s − 1.69·30-s − 0.528·31-s − 4.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.563630929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.563630929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 - 3.38T + 5T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 + 4.43T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 0.669T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 + 8.17T + 67T^{2} \) |
| 71 | \( 1 + 0.295T + 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 + 3.02T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.883763746974079522603249775729, −7.934977818515426038425704972704, −7.43961582566031500237462252498, −6.62281871324167538649637963581, −5.96160662986351753201722615557, −5.13053603591918437776703932186, −3.27012509878601329112118013054, −2.67885461736113436404943327548, −1.64224581593017860357102926229, −1.07850548943618944747679333832,
1.07850548943618944747679333832, 1.64224581593017860357102926229, 2.67885461736113436404943327548, 3.27012509878601329112118013054, 5.13053603591918437776703932186, 5.96160662986351753201722615557, 6.62281871324167538649637963581, 7.43961582566031500237462252498, 7.934977818515426038425704972704, 8.883763746974079522603249775729
