L(s) = 1 | + 2-s − 1.32·3-s + 4-s + 5-s − 1.32·6-s − 7-s + 8-s − 1.25·9-s + 10-s − 1.32·12-s + 5.34·13-s − 14-s − 1.32·15-s + 16-s − 6.44·17-s − 1.25·18-s − 6.42·19-s + 20-s + 1.32·21-s + 5.51·23-s − 1.32·24-s + 25-s + 5.34·26-s + 5.62·27-s − 28-s − 8.19·29-s − 1.32·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.762·3-s + 0.5·4-s + 0.447·5-s − 0.539·6-s − 0.377·7-s + 0.353·8-s − 0.418·9-s + 0.316·10-s − 0.381·12-s + 1.48·13-s − 0.267·14-s − 0.341·15-s + 0.250·16-s − 1.56·17-s − 0.295·18-s − 1.47·19-s + 0.223·20-s + 0.288·21-s + 1.14·23-s − 0.269·24-s + 0.200·25-s + 1.04·26-s + 1.08·27-s − 0.188·28-s − 1.52·29-s − 0.241·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 1.32T + 3T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2.08T + 41T^{2} \) |
| 43 | \( 1 + 2.71T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 5.54T + 73T^{2} \) |
| 79 | \( 1 + 7.35T + 79T^{2} \) |
| 83 | \( 1 + 7.19T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 + 0.548T + 97T^{2} \) |
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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
−7.06267565364705965642747032621, −6.46406774915150690874162506571, −6.04581469702431695364915540155, −5.56845076375964506269095173617, −4.56097764827422491705912452208, −4.12263118232089893232358714677, −3.08033679931687761788169513508, −2.35137843060326188664069937509, −1.30973733228809207115518102709, 0,
1.30973733228809207115518102709, 2.35137843060326188664069937509, 3.08033679931687761788169513508, 4.12263118232089893232358714677, 4.56097764827422491705912452208, 5.56845076375964506269095173617, 6.04581469702431695364915540155, 6.46406774915150690874162506571, 7.06267565364705965642747032621