L(s) = 1 | + 2.74·2-s − 3-s + 5.55·4-s + 0.373·5-s − 2.74·6-s − 2.56·7-s + 9.76·8-s + 9-s + 1.02·10-s + 5.33·11-s − 5.55·12-s + 1.62·13-s − 7.05·14-s − 0.373·15-s + 15.7·16-s − 17-s + 2.74·18-s + 6.40·19-s + 2.07·20-s + 2.56·21-s + 14.6·22-s − 8.29·23-s − 9.76·24-s − 4.86·25-s + 4.47·26-s − 27-s − 14.2·28-s + ⋯ |
L(s) = 1 | + 1.94·2-s − 0.577·3-s + 2.77·4-s + 0.167·5-s − 1.12·6-s − 0.970·7-s + 3.45·8-s + 0.333·9-s + 0.324·10-s + 1.60·11-s − 1.60·12-s + 0.451·13-s − 1.88·14-s − 0.0965·15-s + 3.93·16-s − 0.242·17-s + 0.647·18-s + 1.46·19-s + 0.464·20-s + 0.560·21-s + 3.12·22-s − 1.72·23-s − 1.99·24-s − 0.972·25-s + 0.877·26-s − 0.192·27-s − 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.399388287\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.399388287\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 5 | \( 1 - 0.373T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 19 | \( 1 - 6.40T + 19T^{2} \) |
| 23 | \( 1 + 8.29T + 23T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 + 0.319T + 31T^{2} \) |
| 37 | \( 1 + 9.68T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 2.22T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 - 0.149T + 53T^{2} \) |
| 59 | \( 1 - 0.248T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 9.62T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.141802131127538057807583432408, −7.14966755031088035396126077165, −6.59189244593073322585987356215, −6.09225486916085199874215698158, −5.57611466510523236900675656044, −4.61654251526713957398100495353, −3.78979192157965765833063298313, −3.47105572823660810944865511207, −2.25957639483167927629321039514, −1.23121136207823174344314831860,
1.23121136207823174344314831860, 2.25957639483167927629321039514, 3.47105572823660810944865511207, 3.78979192157965765833063298313, 4.61654251526713957398100495353, 5.57611466510523236900675656044, 6.09225486916085199874215698158, 6.59189244593073322585987356215, 7.14966755031088035396126077165, 8.141802131127538057807583432408