L(s) = 1 | + 2-s − 2.59·3-s + 4-s + 4.27·5-s − 2.59·6-s + 3.34·7-s + 8-s + 3.75·9-s + 4.27·10-s + 0.315·11-s − 2.59·12-s − 6.59·13-s + 3.34·14-s − 11.1·15-s + 16-s + 3.89·17-s + 3.75·18-s − 1.16·19-s + 4.27·20-s − 8.68·21-s + 0.315·22-s + 0.687·23-s − 2.59·24-s + 13.3·25-s − 6.59·26-s − 1.95·27-s + 3.34·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.50·3-s + 0.5·4-s + 1.91·5-s − 1.06·6-s + 1.26·7-s + 0.353·8-s + 1.25·9-s + 1.35·10-s + 0.0952·11-s − 0.750·12-s − 1.82·13-s + 0.893·14-s − 2.87·15-s + 0.250·16-s + 0.944·17-s + 0.884·18-s − 0.266·19-s + 0.956·20-s − 1.89·21-s + 0.0673·22-s + 0.143·23-s − 0.530·24-s + 2.66·25-s − 1.29·26-s − 0.375·27-s + 0.631·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.082408446\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.082408446\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 - 4.27T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 - 0.315T + 11T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 0.687T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 - 4.48T + 31T^{2} \) |
| 37 | \( 1 + 8.76T + 37T^{2} \) |
| 41 | \( 1 - 1.63T + 41T^{2} \) |
| 43 | \( 1 - 4.62T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 + 0.422T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 - 8.94T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 - 0.442T + 73T^{2} \) |
| 79 | \( 1 + 4.06T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 0.114T + 89T^{2} \) |
| 97 | \( 1 - 2.93T + 97T^{2} \) |
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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
−8.377499467956330034523937182270, −7.27570096746169423017786187308, −6.79043490896744448085585494600, −5.88334675439778935424805009698, −5.37257435952036222051561628581, −5.08575387643494834237743040268, −4.35387844711930767326740740873, −2.68959718387444897625310867507, −1.95217614883726726484143673096, −1.04156979669329782725318032271,
1.04156979669329782725318032271, 1.95217614883726726484143673096, 2.68959718387444897625310867507, 4.35387844711930767326740740873, 5.08575387643494834237743040268, 5.37257435952036222051561628581, 5.88334675439778935424805009698, 6.79043490896744448085585494600, 7.27570096746169423017786187308, 8.377499467956330034523937182270