| L(s) = 1 | − i·3-s + 2i·7-s + 2·9-s + i·13-s − 6i·17-s − 2·19-s + 2·21-s + i·23-s − 5i·27-s + 3·29-s + 5·31-s + 8i·37-s + 39-s + 3·41-s − 8i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.755i·7-s + 0.666·9-s + 0.277i·13-s − 1.45i·17-s − 0.458·19-s + 0.436·21-s + 0.208i·23-s − 0.962i·27-s + 0.557·29-s + 0.898·31-s + 1.31i·37-s + 0.160·39-s + 0.468·41-s − 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909254951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909254951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.831456210107175007266278866168, −8.235206556170506111986937114080, −7.24712386481719225679223675556, −6.78419227594162703006681007757, −5.89607936875442353457127825616, −4.98323577386822021303552664044, −4.19723181045914320307516262559, −2.90009683331740490879665127060, −2.12199275246863832848412938979, −0.882161353392458218712659694966,
0.968204168022438439367862263148, 2.23772419551646112198560166838, 3.61545645281537600779155628910, 4.12769631603432109831009053895, 4.91276884538335403575254288265, 5.97357821525095593476757098175, 6.73709704122140940156918410691, 7.55898345363030104854157684122, 8.314211003121809836656248374167, 9.087116303443760518142614216280
