| L(s) = 1 | + 7-s − 3·11-s + 2·13-s + 5·19-s + 23-s − 5·25-s − 6·29-s − 10·31-s + 2·37-s − 3·41-s − 4·43-s − 3·47-s + 49-s − 3·53-s − 9·59-s − 61-s + 8·67-s + 12·71-s + 8·73-s − 3·77-s + 14·79-s − 6·83-s + 6·89-s + 2·91-s − 10·97-s − 15·101-s − 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.554·13-s + 1.14·19-s + 0.208·23-s − 25-s − 1.11·29-s − 1.79·31-s + 0.328·37-s − 0.468·41-s − 0.609·43-s − 0.437·47-s + 1/7·49-s − 0.412·53-s − 1.17·59-s − 0.128·61-s + 0.977·67-s + 1.42·71-s + 0.936·73-s − 0.341·77-s + 1.57·79-s − 0.658·83-s + 0.635·89-s + 0.209·91-s − 1.01·97-s − 1.49·101-s − 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.82488286036153902205964395059, −7.18814632933995790364040467301, −6.29902201220350191733517348523, −5.37377861109351366521937666972, −5.15255859608970007438965620917, −3.92107040789395054491135860985, −3.35236939238212511602577634943, −2.26906912720472421740666543995, −1.40864989317128594336379846250, 0,
1.40864989317128594336379846250, 2.26906912720472421740666543995, 3.35236939238212511602577634943, 3.92107040789395054491135860985, 5.15255859608970007438965620917, 5.37377861109351366521937666972, 6.29902201220350191733517348523, 7.18814632933995790364040467301, 7.82488286036153902205964395059
