| L(s) = 1 | + 2-s + 3.09·3-s + 4-s − 5-s + 3.09·6-s + 2.42·7-s + 8-s + 6.56·9-s − 10-s − 11-s + 3.09·12-s + 2.41·13-s + 2.42·14-s − 3.09·15-s + 16-s + 1.76·17-s + 6.56·18-s + 2.28·19-s − 20-s + 7.50·21-s − 22-s − 2.81·23-s + 3.09·24-s + 25-s + 2.41·26-s + 11.0·27-s + 2.42·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.78·3-s + 0.5·4-s − 0.447·5-s + 1.26·6-s + 0.916·7-s + 0.353·8-s + 2.18·9-s − 0.316·10-s − 0.301·11-s + 0.892·12-s + 0.669·13-s + 0.648·14-s − 0.798·15-s + 0.250·16-s + 0.428·17-s + 1.54·18-s + 0.524·19-s − 0.223·20-s + 1.63·21-s − 0.213·22-s − 0.587·23-s + 0.631·24-s + 0.200·25-s + 0.473·26-s + 2.12·27-s + 0.458·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.600811129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.600811129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 + 2.81T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 7.57T + 37T^{2} \) |
| 41 | \( 1 + 9.99T + 41T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 + 6.69T + 59T^{2} \) |
| 61 | \( 1 - 7.66T + 61T^{2} \) |
| 67 | \( 1 + 8.13T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 8.00T + 73T^{2} \) |
| 79 | \( 1 - 0.188T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
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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
−8.131167648115914809935324197131, −7.70829451463759623878128949555, −7.18153280548083632577405284188, −6.08153594021538742722866448982, −5.11935577997321367609178875124, −4.32939653752698316968056954331, −3.66552259822066565555505657336, −3.06978824823047874637458673421, −2.12513604461184602846713409886, −1.36788756072564474072621428665,
1.36788756072564474072621428665, 2.12513604461184602846713409886, 3.06978824823047874637458673421, 3.66552259822066565555505657336, 4.32939653752698316968056954331, 5.11935577997321367609178875124, 6.08153594021538742722866448982, 7.18153280548083632577405284188, 7.70829451463759623878128949555, 8.131167648115914809935324197131
