| L(s) = 1 | + 2·7-s − 13-s − 2·19-s + 6·23-s + 4·31-s − 2·37-s + 6·41-s − 4·43-s − 3·49-s − 6·53-s − 10·61-s + 8·67-s − 8·73-s − 8·79-s + 12·83-s − 6·89-s − 2·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.277·13-s − 0.458·19-s + 1.25·23-s + 0.718·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 3/7·49-s − 0.824·53-s − 1.28·61-s + 0.977·67-s − 0.936·73-s − 0.900·79-s + 1.31·83-s − 0.635·89-s − 0.209·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.84334202281759, −14.38833467503488, −13.91309406429283, −13.25413268723449, −12.86668064852781, −12.24012868889411, −11.77592932377081, −11.14101988752644, −10.84451851896193, −10.22565831955995, −9.614620458174231, −9.046017524768265, −8.569696777836572, −7.918850623278157, −7.567025822484625, −6.781533493202690, −6.395143672398712, −5.595817004095766, −5.003155543493424, −4.586460236248655, −3.925306257943858, −3.084089814753444, −2.551398562476162, −1.693011657483258, −1.079393262320527, 0,
1.079393262320527, 1.693011657483258, 2.551398562476162, 3.084089814753444, 3.925306257943858, 4.586460236248655, 5.003155543493424, 5.595817004095766, 6.395143672398712, 6.781533493202690, 7.567025822484625, 7.918850623278157, 8.569696777836572, 9.046017524768265, 9.614620458174231, 10.22565831955995, 10.84451851896193, 11.14101988752644, 11.77592932377081, 12.24012868889411, 12.86668064852781, 13.25413268723449, 13.91309406429283, 14.38833467503488, 14.84334202281759
