L(s) = 1 | + 2-s − 1.61·3-s + 4-s + 5-s − 1.61·6-s − 0.381·7-s + 8-s − 0.381·9-s + 10-s + 11-s − 1.61·12-s − 4.47·13-s − 0.381·14-s − 1.61·15-s + 16-s + 5.09·17-s − 0.381·18-s − 4.61·19-s + 20-s + 0.618·21-s + 22-s − 4·23-s − 1.61·24-s + 25-s − 4.47·26-s + 5.47·27-s − 0.381·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.934·3-s + 0.5·4-s + 0.447·5-s − 0.660·6-s − 0.144·7-s + 0.353·8-s − 0.127·9-s + 0.316·10-s + 0.301·11-s − 0.467·12-s − 1.24·13-s − 0.102·14-s − 0.417·15-s + 0.250·16-s + 1.23·17-s − 0.0900·18-s − 1.05·19-s + 0.223·20-s + 0.134·21-s + 0.213·22-s − 0.834·23-s − 0.330·24-s + 0.200·25-s − 0.877·26-s + 1.05·27-s − 0.0721·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 + 0.381T + 7T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 5.09T + 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 1.23T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + 7.09T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 + 6.47T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 + 2.56T + 83T^{2} \) |
| 89 | \( 1 + 0.472T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−7.86287338420571640543391143487, −6.81805109552024025042141403564, −6.35835234378701790805656196398, −5.74928886993051714266829159707, −4.96334616561461872727493300501, −4.51349780441761977361342639339, −3.31578958849075413563010231569, −2.54573586922476454734207698199, −1.43506036725246651011004485126, 0,
1.43506036725246651011004485126, 2.54573586922476454734207698199, 3.31578958849075413563010231569, 4.51349780441761977361342639339, 4.96334616561461872727493300501, 5.74928886993051714266829159707, 6.35835234378701790805656196398, 6.81805109552024025042141403564, 7.86287338420571640543391143487