| L(s) = 1 | − 0.697·2-s − 1.51·4-s − 0.302·7-s + 2.45·8-s − 2.71·11-s − 3.96·13-s + 0.210·14-s + 1.31·16-s + 6.05·17-s + 0.452·19-s + 1.89·22-s − 4.37·23-s + 2.76·26-s + 0.457·28-s + 3.48·29-s − 31-s − 5.82·32-s − 4.22·34-s + 1.56·37-s − 0.315·38-s + 9.00·41-s − 8.93·43-s + 4.11·44-s + 3.04·46-s + 6.64·47-s − 6.90·49-s + 5.99·52-s + ⋯ |
| L(s) = 1 | − 0.493·2-s − 0.756·4-s − 0.114·7-s + 0.866·8-s − 0.819·11-s − 1.09·13-s + 0.0563·14-s + 0.328·16-s + 1.46·17-s + 0.103·19-s + 0.404·22-s − 0.911·23-s + 0.542·26-s + 0.0864·28-s + 0.647·29-s − 0.179·31-s − 1.02·32-s − 0.724·34-s + 0.256·37-s − 0.0512·38-s + 1.40·41-s − 1.36·43-s + 0.620·44-s + 0.449·46-s + 0.968·47-s − 0.986·49-s + 0.831·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.697T + 2T^{2} \) |
| 7 | \( 1 + 0.302T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 - 6.05T + 17T^{2} \) |
| 19 | \( 1 - 0.452T + 19T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 - 9.00T + 41T^{2} \) |
| 43 | \( 1 + 8.93T + 43T^{2} \) |
| 47 | \( 1 - 6.64T + 47T^{2} \) |
| 53 | \( 1 - 8.49T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 + 5.35T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−7.73411593482882843096693496368, −7.24042514441482874336596847727, −6.11091798266699092822391609132, −5.37391596886383556996869265746, −4.84271080799843396475488000405, −4.01831926740037930172078190892, −3.13819401915567654538539058667, −2.22522777929900089124657700495, −1.04319692135639875386028276870, 0,
1.04319692135639875386028276870, 2.22522777929900089124657700495, 3.13819401915567654538539058667, 4.01831926740037930172078190892, 4.84271080799843396475488000405, 5.37391596886383556996869265746, 6.11091798266699092822391609132, 7.24042514441482874336596847727, 7.73411593482882843096693496368
