L(s) = 1 | − 5-s + 2·11-s + 13-s + 17-s + 2·19-s − 4·25-s + 3·29-s + 5·31-s + 4·37-s − 7·41-s − 2·43-s − 7·47-s + 6·53-s − 2·55-s − 9·59-s + 6·61-s − 65-s + 16·67-s − 2·71-s + 2·73-s − 16·79-s + 83-s − 85-s − 12·89-s − 2·95-s + 18·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.603·11-s + 0.277·13-s + 0.242·17-s + 0.458·19-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 0.657·37-s − 1.09·41-s − 0.304·43-s − 1.02·47-s + 0.824·53-s − 0.269·55-s − 1.17·59-s + 0.768·61-s − 0.124·65-s + 1.95·67-s − 0.237·71-s + 0.234·73-s − 1.80·79-s + 0.109·83-s − 0.108·85-s − 1.27·89-s − 0.205·95-s + 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.82408797446619, −13.33860693474031, −12.86556936411853, −12.27013390355949, −11.75053823030422, −11.54393331857244, −11.05555564072753, −10.28495527111778, −9.907272048368868, −9.552193546602535, −8.740968897689357, −8.467273818907195, −7.881011210446978, −7.447060667366889, −6.747501309070726, −6.414111265041879, −5.799186419352346, −5.178727169621259, −4.637393669879596, −4.026987151795780, −3.553690246900796, −2.986444328559581, −2.274524790023759, −1.481141510277413, −0.9098882016566036, 0,
0.9098882016566036, 1.481141510277413, 2.274524790023759, 2.986444328559581, 3.553690246900796, 4.026987151795780, 4.637393669879596, 5.178727169621259, 5.799186419352346, 6.414111265041879, 6.747501309070726, 7.447060667366889, 7.881011210446978, 8.467273818907195, 8.740968897689357, 9.552193546602535, 9.907272048368868, 10.28495527111778, 11.05555564072753, 11.54393331857244, 11.75053823030422, 12.27013390355949, 12.86556936411853, 13.33860693474031, 13.82408797446619