L(s) = 1 | − 2i·7-s + 2·11-s + 4i·13-s + 2i·17-s − 4·19-s + 8i·23-s + 10·29-s + 4·31-s − 8i·43-s − 8i·47-s + 3·49-s + 6i·53-s + 14·59-s − 14·61-s + 4i·67-s + ⋯ |
L(s) = 1 | − 0.755i·7-s + 0.603·11-s + 1.10i·13-s + 0.485i·17-s − 0.917·19-s + 1.66i·23-s + 1.85·29-s + 0.718·31-s − 1.21i·43-s − 1.16i·47-s + 0.428·49-s + 0.824i·53-s + 1.82·59-s − 1.79·61-s + 0.488i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.718731213\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.718731213\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.271773419123833385972089908027, −8.619226812958124793251286345901, −7.73853617694970909897406789525, −6.81784711835040972413300510619, −6.38860910606435673084345880205, −5.20280351159204206088009880530, −4.20360961554558155036213553871, −3.67515982335702484541258848757, −2.24005202008682866580988102327, −1.10789751895787327006907272819,
0.77645809331992266648774328512, 2.35527977505076145344492157759, 3.07907867138672676635866812957, 4.38805168579267703072033013695, 5.05696721617575352290537122627, 6.23843115693072089290230569932, 6.53930608845946281290459124813, 7.84925776426093251091374600814, 8.445842465434164654486167249533, 9.092350677771959683822234290490