L(s) = 1 | + 2-s − 0.132·3-s + 4-s + 2.32·5-s − 0.132·6-s + 1.65·7-s + 8-s − 2.98·9-s + 2.32·10-s + 0.845·11-s − 0.132·12-s + 1.65·14-s − 0.308·15-s + 16-s + 4.99·17-s − 2.98·18-s + 19-s + 2.32·20-s − 0.220·21-s + 0.845·22-s − 1.97·23-s − 0.132·24-s + 0.384·25-s + 0.795·27-s + 1.65·28-s + 4.58·29-s − 0.308·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.0767·3-s + 0.5·4-s + 1.03·5-s − 0.0542·6-s + 0.626·7-s + 0.353·8-s − 0.994·9-s + 0.733·10-s + 0.254·11-s − 0.0383·12-s + 0.443·14-s − 0.0796·15-s + 0.250·16-s + 1.21·17-s − 0.702·18-s + 0.229·19-s + 0.518·20-s − 0.0481·21-s + 0.180·22-s − 0.412·23-s − 0.0271·24-s + 0.0768·25-s + 0.153·27-s + 0.313·28-s + 0.850·29-s − 0.0563·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.268421286\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.268421286\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 0.132T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 0.845T + 11T^{2} \) |
| 17 | \( 1 - 4.99T + 17T^{2} \) |
| 23 | \( 1 + 1.97T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 + 1.11T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 0.0249T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 - 6.65T + 59T^{2} \) |
| 61 | \( 1 + 3.14T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 0.258T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 + 7.27T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 4.01T + 97T^{2} \) |
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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
−7.966302450613839779736223501458, −7.25833910251265065733389697249, −6.30607027525585996602319669168, −5.75123418663481053170949662950, −5.37521042100846332352201733960, −4.52670334329386905292006041986, −3.59215553232164308856077032580, −2.74487323433816118227816864331, −2.00796133008174732377546153250, −1.01154495064337160717119706844,
1.01154495064337160717119706844, 2.00796133008174732377546153250, 2.74487323433816118227816864331, 3.59215553232164308856077032580, 4.52670334329386905292006041986, 5.37521042100846332352201733960, 5.75123418663481053170949662950, 6.30607027525585996602319669168, 7.25833910251265065733389697249, 7.966302450613839779736223501458