| L(s) = 1 | − 5-s − 2·13-s + 2·19-s + 25-s − 6·29-s + 8·31-s − 4·37-s + 6·41-s − 2·43-s + 6·47-s − 6·53-s − 12·59-s − 8·61-s + 2·65-s − 2·67-s + 6·71-s − 2·73-s + 16·79-s + 6·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.554·13-s + 0.458·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.657·37-s + 0.937·41-s − 0.304·43-s + 0.875·47-s − 0.824·53-s − 1.56·59-s − 1.02·61-s + 0.248·65-s − 0.244·67-s + 0.712·71-s − 0.234·73-s + 1.80·79-s + 0.635·89-s − 0.205·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−15.14837840549662, −14.81006211813644, −14.05092683053886, −13.74946444620755, −13.09616617368599, −12.46024224988935, −12.04988642347861, −11.64537943790248, −10.82080344978998, −10.63646455208276, −9.773924779038878, −9.343173950670269, −8.849111930370379, −7.982518964130148, −7.732327357039714, −7.124821474195212, −6.434007855953392, −5.899994058281654, −5.106217051112764, −4.659903279472570, −3.951294157789031, −3.283172556686506, −2.650865122883149, −1.844140407087393, −0.9417548779915030, 0,
0.9417548779915030, 1.844140407087393, 2.650865122883149, 3.283172556686506, 3.951294157789031, 4.659903279472570, 5.106217051112764, 5.899994058281654, 6.434007855953392, 7.124821474195212, 7.732327357039714, 7.982518964130148, 8.849111930370379, 9.343173950670269, 9.773924779038878, 10.63646455208276, 10.82080344978998, 11.64537943790248, 12.04988642347861, 12.46024224988935, 13.09616617368599, 13.74946444620755, 14.05092683053886, 14.81006211813644, 15.14837840549662
