| L(s) = 1 | + 1.20·2-s + 0.150·3-s − 0.537·4-s + 3.32·5-s + 0.181·6-s − 3.06·8-s − 2.97·9-s + 4.02·10-s + 2.93·11-s − 0.0807·12-s + 1.58·13-s + 0.500·15-s − 2.63·16-s + 1.28·17-s − 3.60·18-s + 2.64·19-s − 1.78·20-s + 3.55·22-s + 7.50·23-s − 0.461·24-s + 6.07·25-s + 1.91·26-s − 0.899·27-s − 0.257·29-s + 0.605·30-s − 7.18·31-s + 2.94·32-s + ⋯ |
| L(s) = 1 | + 0.855·2-s + 0.0868·3-s − 0.268·4-s + 1.48·5-s + 0.0742·6-s − 1.08·8-s − 0.992·9-s + 1.27·10-s + 0.886·11-s − 0.0233·12-s + 0.439·13-s + 0.129·15-s − 0.659·16-s + 0.311·17-s − 0.848·18-s + 0.605·19-s − 0.399·20-s + 0.758·22-s + 1.56·23-s − 0.0942·24-s + 1.21·25-s + 0.375·26-s − 0.173·27-s − 0.0478·29-s + 0.110·30-s − 1.29·31-s + 0.520·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.183702963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.183702963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.20T + 2T^{2} \) |
| 3 | \( 1 - 0.150T + 3T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 1.28T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 + 0.257T + 29T^{2} \) |
| 31 | \( 1 + 7.18T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 6.89T + 53T^{2} \) |
| 59 | \( 1 + 9.78T + 59T^{2} \) |
| 61 | \( 1 - 4.88T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 + 9.76T + 71T^{2} \) |
| 73 | \( 1 + 4.95T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 7.63T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
| 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
−9.007764021757450858854974754383, −8.874874071211370962519888623314, −7.43021857850129177824448450577, −6.40358260116887874635313449844, −5.71530400309823902392005234172, −5.40626036734078558623897356462, −4.28902029452143183433592676870, −3.27744638974964220798193445185, −2.51405401684948792278037275875, −1.12653453196731219597957967265,
1.12653453196731219597957967265, 2.51405401684948792278037275875, 3.27744638974964220798193445185, 4.28902029452143183433592676870, 5.40626036734078558623897356462, 5.71530400309823902392005234172, 6.40358260116887874635313449844, 7.43021857850129177824448450577, 8.874874071211370962519888623314, 9.007764021757450858854974754383
