L(s) = 1 | + 0.475·2-s − 3-s − 1.77·4-s − 5-s − 0.475·6-s − 1.79·8-s + 9-s − 0.475·10-s − 11-s + 1.77·12-s − 6.31·13-s + 15-s + 2.69·16-s + 1.92·17-s + 0.475·18-s − 3.92·19-s + 1.77·20-s − 0.475·22-s + 4.74·23-s + 1.79·24-s + 25-s − 3.00·26-s − 27-s − 4.08·29-s + 0.475·30-s + 7.20·31-s + 4.87·32-s + ⋯ |
L(s) = 1 | + 0.336·2-s − 0.577·3-s − 0.886·4-s − 0.447·5-s − 0.194·6-s − 0.634·8-s + 0.333·9-s − 0.150·10-s − 0.301·11-s + 0.511·12-s − 1.75·13-s + 0.258·15-s + 0.673·16-s + 0.466·17-s + 0.112·18-s − 0.899·19-s + 0.396·20-s − 0.101·22-s + 0.988·23-s + 0.366·24-s + 0.200·25-s − 0.589·26-s − 0.192·27-s − 0.757·29-s + 0.0868·30-s + 1.29·31-s + 0.861·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 0.475T + 2T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 + 4.77T + 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 61 | \( 1 + 5.93T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 6.01T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 + 1.69T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.61917157123694959675384676139, −6.74224486618789505639980639119, −5.92117581741750983260494751369, −5.28818929271623333435258197385, −4.53986017273306239359107620945, −4.30139520692388679135707403443, −3.14732566190392571119772344088, −2.42438102249079100221900269008, −0.920953741328327670968018603231, 0,
0.920953741328327670968018603231, 2.42438102249079100221900269008, 3.14732566190392571119772344088, 4.30139520692388679135707403443, 4.53986017273306239359107620945, 5.28818929271623333435258197385, 5.92117581741750983260494751369, 6.74224486618789505639980639119, 7.61917157123694959675384676139