| L(s) = 1 | + 0.117·2-s − 1.63·3-s − 1.98·4-s − 2.28·5-s − 0.192·6-s − 0.468·8-s − 0.325·9-s − 0.269·10-s − 3.49·11-s + 3.24·12-s + 0.309·13-s + 3.74·15-s + 3.91·16-s + 1.81·17-s − 0.0382·18-s − 7.89·19-s + 4.54·20-s − 0.410·22-s − 2.28·23-s + 0.766·24-s + 0.233·25-s + 0.0364·26-s + 5.43·27-s + 5.11·29-s + 0.440·30-s + 6.98·31-s + 1.39·32-s + ⋯ |
| L(s) = 1 | + 0.0831·2-s − 0.944·3-s − 0.993·4-s − 1.02·5-s − 0.0785·6-s − 0.165·8-s − 0.108·9-s − 0.0850·10-s − 1.05·11-s + 0.937·12-s + 0.0859·13-s + 0.965·15-s + 0.979·16-s + 0.439·17-s − 0.00902·18-s − 1.81·19-s + 1.01·20-s − 0.0876·22-s − 0.476·23-s + 0.156·24-s + 0.0466·25-s + 0.00714·26-s + 1.04·27-s + 0.949·29-s + 0.0803·30-s + 1.25·31-s + 0.247·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.117T + 2T^{2} \) |
| 3 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 - 0.309T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 + 7.89T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 + 8.44T + 37T^{2} \) |
| 41 | \( 1 - 7.91T + 41T^{2} \) |
| 43 | \( 1 + 1.00T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 53 | \( 1 - 4.42T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 + 2.05T + 71T^{2} \) |
| 73 | \( 1 + 1.56T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 3.56T + 83T^{2} \) |
| 89 | \( 1 - 3.06T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.006943799076774172823956933975, −6.88656984603684478962240459763, −6.17208399956251288295431078158, −5.43673345818659003594033962034, −4.80856261306876379462347746268, −4.21653706726582034782060224543, −3.44484513458091150663029158045, −2.40674160213922615348569601633, −0.75215254083349391763656613647, 0,
0.75215254083349391763656613647, 2.40674160213922615348569601633, 3.44484513458091150663029158045, 4.21653706726582034782060224543, 4.80856261306876379462347746268, 5.43673345818659003594033962034, 6.17208399956251288295431078158, 6.88656984603684478962240459763, 8.006943799076774172823956933975
