| L(s) = 1 | + 1.13·2-s + 0.0976·3-s − 0.715·4-s − 3.57·5-s + 0.110·6-s + 0.149·7-s − 3.07·8-s − 2.99·9-s − 4.04·10-s − 0.0698·12-s + 13-s + 0.169·14-s − 0.348·15-s − 2.05·16-s + 5.85·17-s − 3.38·18-s + 7.07·19-s + 2.55·20-s + 0.0145·21-s − 1.14·23-s − 0.300·24-s + 7.74·25-s + 1.13·26-s − 0.584·27-s − 0.106·28-s − 3.81·29-s − 0.395·30-s + ⋯ |
| L(s) = 1 | + 0.801·2-s + 0.0563·3-s − 0.357·4-s − 1.59·5-s + 0.0451·6-s + 0.0563·7-s − 1.08·8-s − 0.996·9-s − 1.27·10-s − 0.0201·12-s + 0.277·13-s + 0.0451·14-s − 0.0899·15-s − 0.514·16-s + 1.42·17-s − 0.798·18-s + 1.62·19-s + 0.571·20-s + 0.00317·21-s − 0.239·23-s − 0.0613·24-s + 1.54·25-s + 0.222·26-s − 0.112·27-s − 0.0201·28-s − 0.708·29-s − 0.0721·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.302938948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302938948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 0.0976T + 3T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 7 | \( 1 - 0.149T + 7T^{2} \) |
| 17 | \( 1 - 5.85T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 + 0.0330T + 31T^{2} \) |
| 37 | \( 1 + 7.44T + 37T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 + 6.22T + 67T^{2} \) |
| 71 | \( 1 + 8.68T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.91T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 5.32T + 97T^{2} \) |
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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
−9.234426369988239158176716944532, −8.556178542510422336205298428914, −7.80722483940499003261238407911, −7.18207351726275776675055772958, −5.73807819359794315591609073630, −5.36956526974537360766141953016, −4.21609751526200308322962338590, −3.53538648389772918536552954586, −2.94978466509147271119344768697, −0.70401898248541361235705096618,
0.70401898248541361235705096618, 2.94978466509147271119344768697, 3.53538648389772918536552954586, 4.21609751526200308322962338590, 5.36956526974537360766141953016, 5.73807819359794315591609073630, 7.18207351726275776675055772958, 7.80722483940499003261238407911, 8.556178542510422336205298428914, 9.234426369988239158176716944532
