L(s) = 1 | − 1.82·2-s + 0.572·3-s + 1.33·4-s + 5-s − 1.04·6-s − 3.97·7-s + 1.21·8-s − 2.67·9-s − 1.82·10-s + 11-s + 0.762·12-s − 5.95·13-s + 7.26·14-s + 0.572·15-s − 4.88·16-s + 0.0624·17-s + 4.87·18-s + 7.47·19-s + 1.33·20-s − 2.27·21-s − 1.82·22-s − 4.78·23-s + 0.698·24-s + 25-s + 10.8·26-s − 3.24·27-s − 5.29·28-s + ⋯ |
L(s) = 1 | − 1.29·2-s + 0.330·3-s + 0.666·4-s + 0.447·5-s − 0.426·6-s − 1.50·7-s + 0.431·8-s − 0.890·9-s − 0.577·10-s + 0.301·11-s + 0.220·12-s − 1.65·13-s + 1.94·14-s + 0.147·15-s − 1.22·16-s + 0.0151·17-s + 1.14·18-s + 1.71·19-s + 0.297·20-s − 0.496·21-s − 0.389·22-s − 0.997·23-s + 0.142·24-s + 0.200·25-s + 2.13·26-s − 0.624·27-s − 1.00·28-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
\(L(1)\) |
\(\approx\) |
\(0.4474637518\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4474637518\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 - 0.572T + 3T^{2} \) |
| 7 | \( 1 + 3.97T + 7T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 - 0.0624T + 17T^{2} \) |
| 19 | \( 1 - 7.47T + 19T^{2} \) |
| 23 | \( 1 + 4.78T + 23T^{2} \) |
| 29 | \( 1 - 0.732T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 + 0.257T + 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 - 6.77T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 - 4.36T + 83T^{2} \) |
| 89 | \( 1 + 7.39T + 89T^{2} \) |
| 97 | \( 1 + 6.04T + 97T^{2} \) |
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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
−8.608921389679276695066371762287, −7.80878152021425302052567030251, −7.17626711709780701352216216326, −6.51014543659030960854084146169, −5.62293471580541313513812521491, −4.79013065826204228480374466325, −3.46328132194273898764325361326, −2.80502733781917098767417993538, −1.87000784452086517936768003791, −0.43733790015470598813474193925,
0.43733790015470598813474193925, 1.87000784452086517936768003791, 2.80502733781917098767417993538, 3.46328132194273898764325361326, 4.79013065826204228480374466325, 5.62293471580541313513812521491, 6.51014543659030960854084146169, 7.17626711709780701352216216326, 7.80878152021425302052567030251, 8.608921389679276695066371762287