| L(s) = 1 | − 4·7-s + 2·13-s + 6·17-s − 4·19-s − 6·29-s − 8·31-s + 2·37-s + 6·41-s + 4·43-s + 9·49-s + 6·53-s + 10·61-s + 4·67-s − 2·73-s − 8·79-s + 12·83-s − 18·89-s − 8·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 24·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.824·53-s + 1.28·61-s + 0.488·67-s − 0.234·73-s − 0.900·79-s + 1.31·83-s − 1.90·89-s − 0.838·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 2.20·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.28839810493300, −16.05065129345175, −15.30839682467418, −14.64453712880203, −14.31042217946360, −13.36969759702698, −13.03352396612403, −12.56651447374081, −12.05875053465977, −11.16846312452313, −10.75427426446881, −9.981795991377771, −9.604292489562196, −9.007289278990736, −8.377174523222303, −7.526794907693353, −7.097634452433220, −6.281271162832570, −5.821266103107618, −5.279388333746701, −4.024825326278763, −3.723994701877763, −2.966575692451078, −2.147399758773100, −1.033610003952489, 0,
1.033610003952489, 2.147399758773100, 2.966575692451078, 3.723994701877763, 4.024825326278763, 5.279388333746701, 5.821266103107618, 6.281271162832570, 7.097634452433220, 7.526794907693353, 8.377174523222303, 9.007289278990736, 9.604292489562196, 9.981795991377771, 10.75427426446881, 11.16846312452313, 12.05875053465977, 12.56651447374081, 13.03352396612403, 13.36969759702698, 14.31042217946360, 14.64453712880203, 15.30839682467418, 16.05065129345175, 16.28839810493300
