| L(s) = 1 | − 3.16·3-s + 2.82·5-s + 7.00·9-s + 4.47·11-s − 8.94·15-s + 4.24·17-s − 3.16·19-s + 8.94·23-s + 3.00·25-s − 12.6·27-s − 6·29-s − 6.32·31-s − 14.1·33-s + 2·37-s + 1.41·41-s − 4.47·43-s + 19.7·45-s + 6.32·47-s − 13.4·51-s + 6·53-s + 12.6·55-s + 10.0·57-s − 3.16·59-s − 8.48·61-s − 28.2·69-s + 8.94·71-s − 7.07·73-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 1.26·5-s + 2.33·9-s + 1.34·11-s − 2.30·15-s + 1.02·17-s − 0.725·19-s + 1.86·23-s + 0.600·25-s − 2.43·27-s − 1.11·29-s − 1.13·31-s − 2.46·33-s + 0.328·37-s + 0.220·41-s − 0.681·43-s + 2.95·45-s + 0.922·47-s − 1.87·51-s + 0.824·53-s + 1.70·55-s + 1.32·57-s − 0.411·59-s − 1.08·61-s − 3.40·69-s + 1.06·71-s − 0.827·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.331604622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331604622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−9.478041682445157585632421273930, −9.068119393144075514121892182386, −7.43597839660723218083970739282, −6.72865583257758003258742984733, −6.05575118982121509425576415061, −5.51586444280384886347712282742, −4.75458039980536393059583826961, −3.64013823846510291786936056699, −1.85893755008855497727091090387, −0.962050876203161734055370874216,
0.962050876203161734055370874216, 1.85893755008855497727091090387, 3.64013823846510291786936056699, 4.75458039980536393059583826961, 5.51586444280384886347712282742, 6.05575118982121509425576415061, 6.72865583257758003258742984733, 7.43597839660723218083970739282, 9.068119393144075514121892182386, 9.478041682445157585632421273930
