| L(s) = 1 | − 2·5-s − 2·7-s + 2·11-s − 2·13-s + 4·19-s + 2·23-s − 25-s − 2·29-s + 10·31-s + 4·35-s + 10·37-s + 2·41-s + 4·43-s − 3·49-s + 6·53-s − 4·55-s − 4·59-s + 2·61-s + 4·65-s − 16·67-s − 10·71-s + 6·73-s − 4·77-s + 6·79-s + 4·83-s − 6·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.603·11-s − 0.554·13-s + 0.917·19-s + 0.417·23-s − 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.676·35-s + 1.64·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 1.95·67-s − 1.18·71-s + 0.702·73-s − 0.455·77-s + 0.675·79-s + 0.439·83-s − 0.635·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.34341155277850, −13.11626533274017, −12.32550240400651, −12.09366908878762, −11.61149472432002, −11.30885509433199, −10.60369624410222, −10.09819889067069, −9.617036436150312, −9.232219878514801, −8.774333144709383, −7.966851803375076, −7.738838819383100, −7.251548827599449, −6.648243141677336, −6.199107250217670, −5.707462970302057, −4.954859609233298, −4.425632245214757, −4.014299038193775, −3.376479363790457, −2.869872863734947, −2.388305418569868, −1.332013091892312, −0.7850647466475851, 0,
0.7850647466475851, 1.332013091892312, 2.388305418569868, 2.869872863734947, 3.376479363790457, 4.014299038193775, 4.425632245214757, 4.954859609233298, 5.707462970302057, 6.199107250217670, 6.648243141677336, 7.251548827599449, 7.738838819383100, 7.966851803375076, 8.774333144709383, 9.232219878514801, 9.617036436150312, 10.09819889067069, 10.60369624410222, 11.30885509433199, 11.61149472432002, 12.09366908878762, 12.32550240400651, 13.11626533274017, 13.34341155277850
