| L(s) = 1 | + 2.72·2-s − 3-s + 5.42·4-s − 1.02·5-s − 2.72·6-s + 4.32·7-s + 9.33·8-s + 9-s − 2.79·10-s + 2.38·11-s − 5.42·12-s + 0.754·13-s + 11.7·14-s + 1.02·15-s + 14.5·16-s − 1.03·17-s + 2.72·18-s − 8.17·19-s − 5.55·20-s − 4.32·21-s + 6.49·22-s + 23-s − 9.33·24-s − 3.94·25-s + 2.05·26-s − 27-s + 23.4·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.577·3-s + 2.71·4-s − 0.458·5-s − 1.11·6-s + 1.63·7-s + 3.29·8-s + 0.333·9-s − 0.883·10-s + 0.718·11-s − 1.56·12-s + 0.209·13-s + 3.15·14-s + 0.264·15-s + 3.64·16-s − 0.252·17-s + 0.642·18-s − 1.87·19-s − 1.24·20-s − 0.944·21-s + 1.38·22-s + 0.208·23-s − 1.90·24-s − 0.789·25-s + 0.403·26-s − 0.192·27-s + 4.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.776033682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.776033682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 5 | \( 1 + 1.02T + 5T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 - 0.754T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 - 1.01T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 - 8.08T + 47T^{2} \) |
| 53 | \( 1 + 0.478T + 53T^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 0.830T + 67T^{2} \) |
| 71 | \( 1 + 1.79T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 2.46T + 79T^{2} \) |
| 83 | \( 1 - 4.46T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 + 0.684T + 97T^{2} \) |
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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
−9.046607629624092658647119232470, −7.936120284559702392556682358936, −7.39154711788446752787739912729, −6.41957118909490207052907037239, −5.83364489933189713846973421918, −4.95015460748061006234223610975, −4.26308054188449638046892759119, −3.88519814254115937014001418652, −2.34331598376472358979606254813, −1.52322863251330693470726400291,
1.52322863251330693470726400291, 2.34331598376472358979606254813, 3.88519814254115937014001418652, 4.26308054188449638046892759119, 4.95015460748061006234223610975, 5.83364489933189713846973421918, 6.41957118909490207052907037239, 7.39154711788446752787739912729, 7.936120284559702392556682358936, 9.046607629624092658647119232470
