L(s) = 1 | − 2.75·2-s − 2.97·3-s + 5.59·4-s − 5-s + 8.20·6-s + 1.32·7-s − 9.92·8-s + 5.85·9-s + 2.75·10-s − 11-s − 16.6·12-s − 5.11·13-s − 3.66·14-s + 2.97·15-s + 16.1·16-s − 2.27·17-s − 16.1·18-s + 3.90·19-s − 5.59·20-s − 3.95·21-s + 2.75·22-s + 5.09·23-s + 29.5·24-s + 25-s + 14.0·26-s − 8.48·27-s + 7.43·28-s + ⋯ |
L(s) = 1 | − 1.94·2-s − 1.71·3-s + 2.79·4-s − 0.447·5-s + 3.34·6-s + 0.502·7-s − 3.50·8-s + 1.95·9-s + 0.871·10-s − 0.301·11-s − 4.80·12-s − 1.41·13-s − 0.978·14-s + 0.768·15-s + 4.03·16-s − 0.552·17-s − 3.80·18-s + 0.896·19-s − 1.25·20-s − 0.862·21-s + 0.587·22-s + 1.06·23-s + 6.02·24-s + 0.200·25-s + 2.76·26-s − 1.63·27-s + 1.40·28-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 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 + 2.97T + 3T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 - 3.90T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 0.305T + 29T^{2} \) |
| 31 | \( 1 + 9.15T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 1.52T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 + 6.04T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 8.14T + 71T^{2} \) |
| 79 | \( 1 - 4.69T + 79T^{2} \) |
| 83 | \( 1 - 0.0678T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 9.01T + 97T^{2} \) |
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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
−7.86899498539866087237654010584, −7.41324130567231324512996391235, −6.94802297347745284910271583133, −6.09964191611276742976028535816, −5.32885948640747938441656223976, −4.58814881963789977920928908895, −3.01372933749609389501162819822, −1.89624394405605100550794001451, −0.867265944268399393876535724420, 0,
0.867265944268399393876535724420, 1.89624394405605100550794001451, 3.01372933749609389501162819822, 4.58814881963789977920928908895, 5.32885948640747938441656223976, 6.09964191611276742976028535816, 6.94802297347745284910271583133, 7.41324130567231324512996391235, 7.86899498539866087237654010584