L(s) = 1 | − 5-s + 11-s − 4·13-s − 3·17-s + 5·19-s − 3·23-s + 25-s + 3·29-s − 4·31-s + 11·37-s − 6·41-s − 43-s − 3·47-s − 12·53-s − 55-s + 15·59-s − 10·61-s + 4·65-s − 4·67-s + 9·71-s + 2·73-s − 16·79-s + 12·83-s + 3·85-s − 12·89-s − 5·95-s + 17·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s − 0.625·23-s + 1/5·25-s + 0.557·29-s − 0.718·31-s + 1.80·37-s − 0.937·41-s − 0.152·43-s − 0.437·47-s − 1.64·53-s − 0.134·55-s + 1.95·59-s − 1.28·61-s + 0.496·65-s − 0.488·67-s + 1.06·71-s + 0.234·73-s − 1.80·79-s + 1.31·83-s + 0.325·85-s − 1.27·89-s − 0.512·95-s + 1.72·97-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 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.20071655124558, −13.46373694817781, −13.07611781731513, −12.52185220202954, −11.99269495608889, −11.63524420665220, −11.20994277154166, −10.62692337278228, −9.928742805768039, −9.652817252458557, −9.140348694936005, −8.517163953296691, −7.893125888482112, −7.618137722683139, −6.930363331753076, −6.566748305081181, −5.847519709828756, −5.278944727769971, −4.630674091001532, −4.328827656099670, −3.500549156710310, −3.018854142749276, −2.322082932265167, −1.667468792420814, −0.7852024296666054, 0,
0.7852024296666054, 1.667468792420814, 2.322082932265167, 3.018854142749276, 3.500549156710310, 4.328827656099670, 4.630674091001532, 5.278944727769971, 5.847519709828756, 6.566748305081181, 6.930363331753076, 7.618137722683139, 7.893125888482112, 8.517163953296691, 9.140348694936005, 9.652817252458557, 9.928742805768039, 10.62692337278228, 11.20994277154166, 11.63524420665220, 11.99269495608889, 12.52185220202954, 13.07611781731513, 13.46373694817781, 14.20071655124558