| L(s) = 1 | − 5-s − 11-s + 13-s + 6·17-s + 2·19-s + 7·23-s + 25-s − 5·29-s + 4·31-s − 10·37-s + 9·41-s + 3·43-s − 9·47-s + 9·53-s + 55-s − 12·59-s − 65-s + 8·67-s − 2·71-s − 2·73-s + 6·79-s − 4·83-s − 6·85-s + 6·89-s − 2·95-s − 16·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 0.277·13-s + 1.45·17-s + 0.458·19-s + 1.45·23-s + 1/5·25-s − 0.928·29-s + 0.718·31-s − 1.64·37-s + 1.40·41-s + 0.457·43-s − 1.31·47-s + 1.23·53-s + 0.134·55-s − 1.56·59-s − 0.124·65-s + 0.977·67-s − 0.237·71-s − 0.234·73-s + 0.675·79-s − 0.439·83-s − 0.650·85-s + 0.635·89-s − 0.205·95-s − 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.98671345340392, −13.61078151143392, −12.95416539081297, −12.57609604153148, −12.08765057960668, −11.61677024333455, −11.04076552582600, −10.67854228311810, −10.11230477767312, −9.536292474384486, −9.101269839755569, −8.522744032937800, −7.944115096919063, −7.537103949421409, −7.079406663912058, −6.468347082782664, −5.782306191459077, −5.265262779249798, −4.899094847739732, −4.069483462143446, −3.520679878769101, −3.056707253579117, −2.434345118947355, −1.419084404659523, −0.9795200026400095, 0,
0.9795200026400095, 1.419084404659523, 2.434345118947355, 3.056707253579117, 3.520679878769101, 4.069483462143446, 4.899094847739732, 5.265262779249798, 5.782306191459077, 6.468347082782664, 7.079406663912058, 7.537103949421409, 7.944115096919063, 8.522744032937800, 9.101269839755569, 9.536292474384486, 10.11230477767312, 10.67854228311810, 11.04076552582600, 11.61677024333455, 12.08765057960668, 12.57609604153148, 12.95416539081297, 13.61078151143392, 13.98671345340392
