L(s) = 1 | + 2-s − 3.29·3-s + 4-s − 1.58·5-s − 3.29·6-s − 7-s + 8-s + 7.83·9-s − 1.58·10-s − 1.04·11-s − 3.29·12-s + 5.91·13-s − 14-s + 5.21·15-s + 16-s + 6.46·17-s + 7.83·18-s + 1.93·19-s − 1.58·20-s + 3.29·21-s − 1.04·22-s − 6.43·23-s − 3.29·24-s − 2.49·25-s + 5.91·26-s − 15.9·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.90·3-s + 0.5·4-s − 0.708·5-s − 1.34·6-s − 0.377·7-s + 0.353·8-s + 2.61·9-s − 0.500·10-s − 0.314·11-s − 0.950·12-s + 1.64·13-s − 0.267·14-s + 1.34·15-s + 0.250·16-s + 1.56·17-s + 1.84·18-s + 0.443·19-s − 0.354·20-s + 0.718·21-s − 0.222·22-s − 1.34·23-s − 0.671·24-s − 0.498·25-s + 1.15·26-s − 3.06·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.442082252\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.442082252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 + 3.29T + 3T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 + 6.43T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 - 1.35T + 47T^{2} \) |
| 53 | \( 1 + 6.74T + 53T^{2} \) |
| 59 | \( 1 - 2.59T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.77127969909602765617321353550, −7.17169536269700064622073327925, −6.29180921602576754863344345447, −5.91860754445498102722175688075, −5.40328595840622678676310558819, −4.45016405738496192083169804730, −3.93142627079771552086991801505, −3.13638768643457278512479508467, −1.50887714465312172186150531752, −0.67725995881546380058564219029,
0.67725995881546380058564219029, 1.50887714465312172186150531752, 3.13638768643457278512479508467, 3.93142627079771552086991801505, 4.45016405738496192083169804730, 5.40328595840622678676310558819, 5.91860754445498102722175688075, 6.29180921602576754863344345447, 7.17169536269700064622073327925, 7.77127969909602765617321353550