L(s) = 1 | + 0.410·2-s + 3-s − 1.83·4-s − 3.27·5-s + 0.410·6-s + 7-s − 1.57·8-s + 9-s − 1.34·10-s − 1.45·11-s − 1.83·12-s − 0.618·13-s + 0.410·14-s − 3.27·15-s + 3.01·16-s − 4.02·17-s + 0.410·18-s − 4.94·19-s + 5.99·20-s + 21-s − 0.598·22-s + 5.12·23-s − 1.57·24-s + 5.71·25-s − 0.253·26-s + 27-s − 1.83·28-s + ⋯ |
L(s) = 1 | + 0.290·2-s + 0.577·3-s − 0.915·4-s − 1.46·5-s + 0.167·6-s + 0.377·7-s − 0.556·8-s + 0.333·9-s − 0.424·10-s − 0.439·11-s − 0.528·12-s − 0.171·13-s + 0.109·14-s − 0.845·15-s + 0.754·16-s − 0.975·17-s + 0.0967·18-s − 1.13·19-s + 1.34·20-s + 0.218·21-s − 0.127·22-s + 1.06·23-s − 0.321·24-s + 1.14·25-s − 0.0497·26-s + 0.192·27-s − 0.346·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 0.410T + 2T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 9.27T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 8.09T + 37T^{2} \) |
| 41 | \( 1 - 0.372T + 41T^{2} \) |
| 43 | \( 1 + 8.02T + 43T^{2} \) |
| 47 | \( 1 - 7.16T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 + 4.95T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 + 4.75T + 71T^{2} \) |
| 73 | \( 1 - 7.00T + 73T^{2} \) |
| 79 | \( 1 + 5.25T + 79T^{2} \) |
| 83 | \( 1 - 8.62T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 4.62T + 97T^{2} \) |
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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
−7.84493053382680770602366948697, −6.82299163272470375614864567321, −6.21749248680423347109178708344, −4.88814129266036569173997414420, −4.56483749990267261180527534617, −4.13846240919485356632446298509, −3.12961596311898684676118366334, −2.60079056322342246692456288457, −1.05429983363472556680601295749, 0,
1.05429983363472556680601295749, 2.60079056322342246692456288457, 3.12961596311898684676118366334, 4.13846240919485356632446298509, 4.56483749990267261180527534617, 4.88814129266036569173997414420, 6.21749248680423347109178708344, 6.82299163272470375614864567321, 7.84493053382680770602366948697