L(s) = 1 | + 5-s + 5·11-s − 2·13-s − 6·17-s − 2·19-s − 6·23-s − 4·25-s − 3·29-s − 5·31-s − 2·37-s − 8·41-s − 4·43-s − 4·47-s + 9·53-s + 5·55-s + 3·59-s + 12·61-s − 2·65-s + 2·67-s − 8·71-s + 14·73-s + 79-s − 17·83-s − 6·85-s − 18·89-s − 2·95-s − 3·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.50·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s − 1.25·23-s − 4/5·25-s − 0.557·29-s − 0.898·31-s − 0.328·37-s − 1.24·41-s − 0.609·43-s − 0.583·47-s + 1.23·53-s + 0.674·55-s + 0.390·59-s + 1.53·61-s − 0.248·65-s + 0.244·67-s − 0.949·71-s + 1.63·73-s + 0.112·79-s − 1.86·83-s − 0.650·85-s − 1.90·89-s − 0.205·95-s − 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.447057141984321735505606763739, −7.27748567667640003421468379727, −6.70324407684425814938448413209, −6.05708384973673803895725585556, −5.20503618499456454297900514746, −4.19003793978491012852514710258, −3.69245432892618504514991499251, −2.26657467530037071537514849863, −1.68243387112320054086860980692, 0,
1.68243387112320054086860980692, 2.26657467530037071537514849863, 3.69245432892618504514991499251, 4.19003793978491012852514710258, 5.20503618499456454297900514746, 6.05708384973673803895725585556, 6.70324407684425814938448413209, 7.27748567667640003421468379727, 8.447057141984321735505606763739