L(s) = 1 | − 2.62·2-s + 4.86·4-s − 4.12·5-s − 0.639·7-s − 7.51·8-s + 10.8·10-s − 11-s + 3.39·13-s + 1.67·14-s + 9.96·16-s − 2.27·17-s − 3.83·19-s − 20.0·20-s + 2.62·22-s − 0.231·23-s + 12.0·25-s − 8.90·26-s − 3.11·28-s + 5.21·29-s − 6.79·31-s − 11.0·32-s + 5.96·34-s + 2.63·35-s + 0.00835·37-s + 10.0·38-s + 31.0·40-s + 1.25·41-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 2.43·4-s − 1.84·5-s − 0.241·7-s − 2.65·8-s + 3.42·10-s − 0.301·11-s + 0.942·13-s + 0.447·14-s + 2.49·16-s − 0.551·17-s − 0.880·19-s − 4.49·20-s + 0.558·22-s − 0.0483·23-s + 2.40·25-s − 1.74·26-s − 0.588·28-s + 0.968·29-s − 1.21·31-s − 1.95·32-s + 1.02·34-s + 0.446·35-s + 0.00137·37-s + 1.63·38-s + 4.90·40-s + 0.195·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 + 4.12T + 5T^{2} \) |
| 7 | \( 1 + 0.639T + 7T^{2} \) |
| 13 | \( 1 - 3.39T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 + 0.231T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 + 6.79T + 31T^{2} \) |
| 37 | \( 1 - 0.00835T + 37T^{2} \) |
| 41 | \( 1 - 1.25T + 41T^{2} \) |
| 43 | \( 1 + 5.91T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 - 5.78T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 + 2.69T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 + 2.54T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−7.997981005971478926525746887879, −7.20391719147090592798587479507, −6.78761185624580749149708073458, −6.00465993335790033407585995708, −4.70263093384777407085507417833, −3.78214066013968139160109935638, −3.09445841311178676233802808312, −2.03497240557326225494534809853, −0.833933194883374237904638074098, 0,
0.833933194883374237904638074098, 2.03497240557326225494534809853, 3.09445841311178676233802808312, 3.78214066013968139160109935638, 4.70263093384777407085507417833, 6.00465993335790033407585995708, 6.78761185624580749149708073458, 7.20391719147090592798587479507, 7.997981005971478926525746887879