| L(s) = 1 | − 4·7-s + 6·11-s − 2·13-s − 17-s + 4·19-s + 4·31-s + 4·37-s − 6·41-s + 8·43-s + 9·49-s − 6·53-s − 4·61-s + 8·67-s − 2·73-s − 24·77-s − 8·79-s + 6·89-s + 8·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 4·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.80·11-s − 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.718·31-s + 0.657·37-s − 0.937·41-s + 1.21·43-s + 9/7·49-s − 0.824·53-s − 0.512·61-s + 0.977·67-s − 0.234·73-s − 2.73·77-s − 0.900·79-s + 0.635·89-s + 0.838·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.58335055850599, −13.94159065542931, −13.61154636023475, −13.03360779735319, −12.34392613885388, −12.19078262435869, −11.58750687924671, −11.07227207046382, −10.35051021860312, −9.744194209862240, −9.443806593457778, −9.155853347825521, −8.418631644201157, −7.746249447264225, −7.038287719222839, −6.683847821196369, −6.271865359874502, −5.674503199830726, −4.961366223072104, −4.105357296064011, −3.862597999467315, −3.018441364868512, −2.670048502613930, −1.591052018244444, −0.9305271855260058, 0,
0.9305271855260058, 1.591052018244444, 2.670048502613930, 3.018441364868512, 3.862597999467315, 4.105357296064011, 4.961366223072104, 5.674503199830726, 6.271865359874502, 6.683847821196369, 7.038287719222839, 7.746249447264225, 8.418631644201157, 9.155853347825521, 9.443806593457778, 9.744194209862240, 10.35051021860312, 11.07227207046382, 11.58750687924671, 12.19078262435869, 12.34392613885388, 13.03360779735319, 13.61154636023475, 13.94159065542931, 14.58335055850599
