| L(s) = 1 | + 2.59·2-s + 3-s + 4.74·4-s − 5-s + 2.59·6-s + 7-s + 7.11·8-s + 9-s − 2.59·10-s − 1.74·11-s + 4.74·12-s + 5.74·13-s + 2.59·14-s − 15-s + 8.99·16-s + 17-s + 2.59·18-s − 0.514·19-s − 4.74·20-s + 21-s − 4.52·22-s − 8.26·23-s + 7.11·24-s + 25-s + 14.9·26-s + 27-s + 4.74·28-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.577·3-s + 2.37·4-s − 0.447·5-s + 1.05·6-s + 0.377·7-s + 2.51·8-s + 0.333·9-s − 0.821·10-s − 0.524·11-s + 1.36·12-s + 1.59·13-s + 0.693·14-s − 0.258·15-s + 2.24·16-s + 0.242·17-s + 0.611·18-s − 0.117·19-s − 1.06·20-s + 0.218·21-s − 0.963·22-s − 1.72·23-s + 1.45·24-s + 0.200·25-s + 2.92·26-s + 0.192·27-s + 0.895·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.483800098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.483800098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 2.59T + 2T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 - 5.74T + 13T^{2} \) |
| 19 | \( 1 + 0.514T + 19T^{2} \) |
| 23 | \( 1 + 8.26T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 31 | \( 1 + 1.74T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.51T + 59T^{2} \) |
| 61 | \( 1 - 5.96T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 7.74T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 7.99T + 83T^{2} \) |
| 89 | \( 1 - 8.77T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−9.214604665630104221778810203124, −8.069631842531778821122490581293, −7.72945316178797549090564108505, −6.60133766851824750728756065249, −5.91001204956921388817028342599, −5.11469510534396608698352926129, −4.00437861004790193435166845705, −3.75027385637544515760897095756, −2.65140239244998467924407652444, −1.64041601859189390881405630255,
1.64041601859189390881405630255, 2.65140239244998467924407652444, 3.75027385637544515760897095756, 4.00437861004790193435166845705, 5.11469510534396608698352926129, 5.91001204956921388817028342599, 6.60133766851824750728756065249, 7.72945316178797549090564108505, 8.069631842531778821122490581293, 9.214604665630104221778810203124
