| L(s) = 1 | + 7-s − 2·13-s + 2·17-s − 4·19-s − 2·29-s − 4·31-s − 2·37-s + 2·41-s + 8·43-s + 4·47-s + 49-s − 2·53-s + 12·59-s − 14·61-s + 8·67-s − 8·71-s + 2·73-s − 8·79-s + 4·83-s + 18·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.371·29-s − 0.718·31-s − 0.328·37-s + 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.274·53-s + 1.56·59-s − 1.79·61-s + 0.977·67-s − 0.949·71-s + 0.234·73-s − 0.900·79-s + 0.439·83-s + 1.90·89-s − 0.209·91-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.65370166595807, −14.97089092165408, −14.54082368799826, −14.20218052219767, −13.45383312836863, −12.87898652734906, −12.44715223777180, −11.86054971601002, −11.32756480438620, −10.60507536523498, −10.38357273033067, −9.507983810949670, −9.076212519704778, −8.482525356605300, −7.735090035486373, −7.416930702936894, −6.673715039233814, −5.983104805802500, −5.424219227622292, −4.760791266488837, −4.118460609694438, −3.481640083439966, −2.562068842177617, −1.996475021020559, −1.065490377235788, 0,
1.065490377235788, 1.996475021020559, 2.562068842177617, 3.481640083439966, 4.118460609694438, 4.760791266488837, 5.424219227622292, 5.983104805802500, 6.673715039233814, 7.416930702936894, 7.735090035486373, 8.482525356605300, 9.076212519704778, 9.507983810949670, 10.38357273033067, 10.60507536523498, 11.32756480438620, 11.86054971601002, 12.44715223777180, 12.87898652734906, 13.45383312836863, 14.20218052219767, 14.54082368799826, 14.97089092165408, 15.65370166595807
