| L(s) = 1 | − 4·7-s + 2·13-s − 8·19-s − 5·25-s − 4·31-s + 10·37-s − 8·43-s + 9·49-s − 14·61-s + 16·67-s + 10·73-s − 4·79-s − 8·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.554·13-s − 1.83·19-s − 25-s − 0.718·31-s + 1.64·37-s − 1.21·43-s + 9/7·49-s − 1.79·61-s + 1.95·67-s + 1.17·73-s − 0.450·79-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5890343028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5890343028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.83996854316154, −14.18625552947314, −13.54388022774312, −13.12146494807075, −12.78362454197565, −12.29250541847101, −11.62102367791059, −10.97118649448681, −10.60025598040564, −9.916194903033687, −9.478439788236182, −9.065379259256257, −8.261608445682847, −7.957674132709340, −7.027462741819345, −6.550330984728910, −6.165258402907657, −5.646901829489682, −4.790444793366107, −3.924534958167956, −3.767647739551200, −2.847326424467570, −2.295245875559307, −1.413499270663325, −0.2741681790894333,
0.2741681790894333, 1.413499270663325, 2.295245875559307, 2.847326424467570, 3.767647739551200, 3.924534958167956, 4.790444793366107, 5.646901829489682, 6.165258402907657, 6.550330984728910, 7.027462741819345, 7.957674132709340, 8.261608445682847, 9.065379259256257, 9.478439788236182, 9.916194903033687, 10.60025598040564, 10.97118649448681, 11.62102367791059, 12.29250541847101, 12.78362454197565, 13.12146494807075, 13.54388022774312, 14.18625552947314, 14.83996854316154
