L(s) = 1 | + 2·5-s − 2·11-s + 4·13-s + 6·17-s − 8·19-s + 6·23-s − 25-s + 10·29-s − 4·31-s + 6·37-s − 6·41-s + 4·43-s + 8·47-s − 2·53-s − 4·55-s − 4·59-s + 8·61-s + 8·65-s − 8·67-s + 10·71-s − 4·73-s + 4·79-s + 12·83-s + 12·85-s − 14·89-s − 16·95-s − 4·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.603·11-s + 1.10·13-s + 1.45·17-s − 1.83·19-s + 1.25·23-s − 1/5·25-s + 1.85·29-s − 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.274·53-s − 0.539·55-s − 0.520·59-s + 1.02·61-s + 0.992·65-s − 0.977·67-s + 1.18·71-s − 0.468·73-s + 0.450·79-s + 1.31·83-s + 1.30·85-s − 1.48·89-s − 1.64·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.147552421\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.147552421\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.256548656183135742900080484788, −8.547949184009728642471294162092, −7.84924368450563575242009882154, −6.74458088630313517047691069955, −6.05286408547326495643983998396, −5.38005753627782483363978707239, −4.37034644412242782519108217341, −3.25226503882638876614695774905, −2.26546408701208752246456475775, −1.06972475178580767206852925346,
1.06972475178580767206852925346, 2.26546408701208752246456475775, 3.25226503882638876614695774905, 4.37034644412242782519108217341, 5.38005753627782483363978707239, 6.05286408547326495643983998396, 6.74458088630313517047691069955, 7.84924368450563575242009882154, 8.547949184009728642471294162092, 9.256548656183135742900080484788