| L(s) = 1 | − 2.23·2-s + 0.121·3-s + 3.00·4-s − 0.270·6-s + 3.51·7-s − 2.23·8-s − 2.98·9-s + 5.38·11-s + 0.363·12-s + 1.00·13-s − 7.86·14-s − 0.996·16-s − 3.67·17-s + 6.67·18-s + 0.426·21-s − 12.0·22-s − 1.82·23-s − 0.271·24-s − 2.24·26-s − 0.725·27-s + 10.5·28-s + 1.36·29-s − 2.46·31-s + 6.70·32-s + 0.651·33-s + 8.21·34-s − 8.95·36-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.0699·3-s + 1.50·4-s − 0.110·6-s + 1.32·7-s − 0.791·8-s − 0.995·9-s + 1.62·11-s + 0.104·12-s + 0.278·13-s − 2.10·14-s − 0.249·16-s − 0.890·17-s + 1.57·18-s + 0.0930·21-s − 2.56·22-s − 0.380·23-s − 0.0553·24-s − 0.441·26-s − 0.139·27-s + 1.99·28-s + 0.252·29-s − 0.442·31-s + 1.18·32-s + 0.113·33-s + 1.40·34-s − 1.49·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.023887100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023887100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 0.121T + 3T^{2} \) |
| 7 | \( 1 - 3.51T + 7T^{2} \) |
| 11 | \( 1 - 5.38T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 - 1.36T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 - 0.168T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 6.84T + 71T^{2} \) |
| 73 | \( 1 - 3.38T + 73T^{2} \) |
| 79 | \( 1 - 3.06T + 79T^{2} \) |
| 83 | \( 1 - 9.87T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 1.87T + 97T^{2} \) |
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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
−7.87255011479972823934633309892, −7.36322117080877705545900438370, −6.52425570799391039375149596989, −6.00702938412253473076858111058, −4.94413992394064584911941418851, −4.24820353111531618875448137518, −3.28091469392303772883491504707, −2.06463820541450137207153658970, −1.65527633450860016278369360634, −0.64038773658120037742649982745,
0.64038773658120037742649982745, 1.65527633450860016278369360634, 2.06463820541450137207153658970, 3.28091469392303772883491504707, 4.24820353111531618875448137518, 4.94413992394064584911941418851, 6.00702938412253473076858111058, 6.52425570799391039375149596989, 7.36322117080877705545900438370, 7.87255011479972823934633309892
