| L(s) = 1 | + 1.73i·3-s − 5-s + (−2 − 1.73i)7-s − 2.99·9-s − 3.46i·11-s − 1.73i·15-s + 6·17-s + 3.46i·19-s + (2.99 − 3.46i)21-s + 3.46i·23-s + 25-s − 5.19i·27-s + 6.92i·29-s − 3.46i·31-s + 5.99·33-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s − 0.447·5-s + (−0.755 − 0.654i)7-s − 0.999·9-s − 1.04i·11-s − 0.447i·15-s + 1.45·17-s + 0.794i·19-s + (0.654 − 0.755i)21-s + 0.722i·23-s + 0.200·25-s − 0.999i·27-s + 1.28i·29-s − 0.622i·31-s + 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304568991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304568991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−9.480696064539601407731812953031, −8.825883601165449380528531189619, −7.902039734334558043555125042796, −7.21463024851942921108605313917, −5.90498972589107391443652397480, −5.54839783082297697438556053599, −4.20038993135112679129094501772, −3.59652540692593678369843025719, −2.93633057642367625802991181019, −0.858340131635840093266484899711,
0.72206754198545490062980935577, 2.23283319459661454064838374185, 2.97299256042288683656230736871, 4.17110607161746032592721463974, 5.36558660343883412833402128804, 6.08534575734793081410296563567, 6.99531770856166885083132845989, 7.51555577643464559491547712804, 8.368801190515624475181747662042, 9.163571076100335377451718206971
