| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 2·13-s + 4·14-s + 16-s − 4·19-s − 20-s + 25-s + 2·26-s + 4·28-s − 6·29-s − 8·31-s + 32-s − 4·35-s − 2·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s + 9·49-s + 50-s + 2·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.676·35-s − 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 9/7·49-s + 0.141·50-s + 0.277·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.33710350484107, −14.93084989818551, −14.63504078049401, −14.07423139950132, −13.42145125099919, −12.97939894737233, −12.37691764853676, −11.72978122313588, −11.34093469713616, −10.90359102773012, −10.45625279354257, −9.610618551250649, −8.743505261608749, −8.389759028109602, −7.864092183739410, −7.106591554319463, −6.790733154333101, −5.644071550836131, −5.496073758243726, −4.682098758245274, −4.077927821699721, −3.673646414171904, −2.702348932863948, −1.839955897595660, −1.398712384968503, 0,
1.398712384968503, 1.839955897595660, 2.702348932863948, 3.673646414171904, 4.077927821699721, 4.682098758245274, 5.496073758243726, 5.644071550836131, 6.790733154333101, 7.106591554319463, 7.864092183739410, 8.389759028109602, 8.743505261608749, 9.610618551250649, 10.45625279354257, 10.90359102773012, 11.34093469713616, 11.72978122313588, 12.37691764853676, 12.97939894737233, 13.42145125099919, 14.07423139950132, 14.63504078049401, 14.93084989818551, 15.33710350484107
