L(s) = 1 | − 3.02·3-s + 1.31·5-s + 3.08·7-s + 6.14·9-s + 4.77·11-s − 4.41·13-s − 3.98·15-s + 2.25·17-s + 19-s − 9.31·21-s − 9.04·23-s − 3.26·25-s − 9.49·27-s − 5.83·29-s − 4.13·31-s − 14.4·33-s + 4.05·35-s + 8.70·37-s + 13.3·39-s − 7.87·41-s − 8.02·43-s + 8.09·45-s + 4.20·47-s + 2.49·49-s − 6.82·51-s − 53-s + 6.29·55-s + ⋯ |
L(s) = 1 | − 1.74·3-s + 0.589·5-s + 1.16·7-s + 2.04·9-s + 1.43·11-s − 1.22·13-s − 1.02·15-s + 0.547·17-s + 0.229·19-s − 2.03·21-s − 1.88·23-s − 0.652·25-s − 1.82·27-s − 1.08·29-s − 0.743·31-s − 2.51·33-s + 0.686·35-s + 1.43·37-s + 2.13·39-s − 1.23·41-s − 1.22·43-s + 1.20·45-s + 0.614·47-s + 0.355·49-s − 0.956·51-s − 0.137·53-s + 0.848·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 23 | \( 1 + 9.04T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 + 7.87T + 41T^{2} \) |
| 43 | \( 1 + 8.02T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 59 | \( 1 + 7.86T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 + 3.37T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 0.0108T + 97T^{2} \) |
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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
−7.70887133262966390304918763030, −7.38228120017802297656210012800, −6.19989401945997685598684192877, −6.01303969097814171810339394757, −5.10522572040472159741970538303, −4.58411360129998994291675871256, −3.74625390934081686040197819276, −1.94138429756034318077934032283, −1.43594292493437214359994479210, 0,
1.43594292493437214359994479210, 1.94138429756034318077934032283, 3.74625390934081686040197819276, 4.58411360129998994291675871256, 5.10522572040472159741970538303, 6.01303969097814171810339394757, 6.19989401945997685598684192877, 7.38228120017802297656210012800, 7.70887133262966390304918763030