| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s + 13-s + 15-s + 17-s + 6·19-s + 3·21-s − 5·23-s + 25-s + 27-s + 6·29-s + 2·31-s + 3·35-s + 7·37-s + 39-s − 3·41-s + 8·43-s + 45-s − 2·47-s + 2·49-s + 51-s − 53-s + 6·57-s + 15·61-s + 3·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.507·35-s + 1.15·37-s + 0.160·39-s − 0.468·41-s + 1.21·43-s + 0.149·45-s − 0.291·47-s + 2/7·49-s + 0.140·51-s − 0.137·53-s + 0.794·57-s + 1.92·61-s + 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.520419173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.520419173\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.16791600195671, −12.70439899642333, −12.15479627543031, −11.65990165888966, −11.30526896372987, −10.79351347284919, −10.13635850322590, −9.792182916224952, −9.437834510161736, −8.714773938848227, −8.314330574818953, −7.912004051773226, −7.540806612736443, −6.881121036008913, −6.322800588034164, −5.795197638025648, −5.151753164365993, −4.864605508362013, −4.149673896803775, −3.670318759216663, −3.009030567296399, −2.353022354777237, −1.968718865713550, −1.120747110644694, −0.8050094144315121,
0.8050094144315121, 1.120747110644694, 1.968718865713550, 2.353022354777237, 3.009030567296399, 3.670318759216663, 4.149673896803775, 4.864605508362013, 5.151753164365993, 5.795197638025648, 6.322800588034164, 6.881121036008913, 7.540806612736443, 7.912004051773226, 8.314330574818953, 8.714773938848227, 9.437834510161736, 9.792182916224952, 10.13635850322590, 10.79351347284919, 11.30526896372987, 11.65990165888966, 12.15479627543031, 12.70439899642333, 13.16791600195671
