| L(s) = 1 | + 1.12·2-s − 3.21·3-s − 0.727·4-s − 5-s − 3.62·6-s − 4.73·7-s − 3.07·8-s + 7.34·9-s − 1.12·10-s + 11-s + 2.34·12-s − 1.61·13-s − 5.34·14-s + 3.21·15-s − 2.01·16-s − 2.75·17-s + 8.28·18-s + 4.78·19-s + 0.727·20-s + 15.2·21-s + 1.12·22-s + 3.97·23-s + 9.89·24-s + 25-s − 1.81·26-s − 13.9·27-s + 3.44·28-s + ⋯ |
| L(s) = 1 | + 0.797·2-s − 1.85·3-s − 0.363·4-s − 0.447·5-s − 1.48·6-s − 1.79·7-s − 1.08·8-s + 2.44·9-s − 0.356·10-s + 0.301·11-s + 0.675·12-s − 0.447·13-s − 1.42·14-s + 0.830·15-s − 0.503·16-s − 0.667·17-s + 1.95·18-s + 1.09·19-s + 0.162·20-s + 3.32·21-s + 0.240·22-s + 0.829·23-s + 2.02·24-s + 0.200·25-s − 0.356·26-s − 2.69·27-s + 0.651·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 + 8.07T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 - 7.03T + 59T^{2} \) |
| 61 | \( 1 + 7.49T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 + 9.02T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.80355500928152600994234035930, −6.76056338513391158914630835778, −6.52329227275078734797773904517, −5.83340836981713718896943709623, −5.06231372599099885086295275226, −4.48366514737331888385763682443, −3.67200860640919165299510589308, −2.86849528889353551010334049428, −0.863213641617168647396878132132, 0,
0.863213641617168647396878132132, 2.86849528889353551010334049428, 3.67200860640919165299510589308, 4.48366514737331888385763682443, 5.06231372599099885086295275226, 5.83340836981713718896943709623, 6.52329227275078734797773904517, 6.76056338513391158914630835778, 7.80355500928152600994234035930
