| L(s) = 1 | + 2-s + 3.30·3-s + 4-s + 1.88·5-s + 3.30·6-s + 2.41·7-s + 8-s + 7.91·9-s + 1.88·10-s + 3.30·12-s − 3.16·13-s + 2.41·14-s + 6.21·15-s + 16-s − 0.0684·17-s + 7.91·18-s − 19-s + 1.88·20-s + 7.96·21-s − 7.39·23-s + 3.30·24-s − 1.45·25-s − 3.16·26-s + 16.2·27-s + 2.41·28-s − 7.11·29-s + 6.21·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.841·5-s + 1.34·6-s + 0.911·7-s + 0.353·8-s + 2.63·9-s + 0.595·10-s + 0.953·12-s − 0.878·13-s + 0.644·14-s + 1.60·15-s + 0.250·16-s − 0.0165·17-s + 1.86·18-s − 0.229·19-s + 0.420·20-s + 1.73·21-s − 1.54·23-s + 0.674·24-s − 0.291·25-s − 0.620·26-s + 3.12·27-s + 0.455·28-s − 1.32·29-s + 1.13·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.871214298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.871214298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 + 0.0684T + 17T^{2} \) |
| 23 | \( 1 + 7.39T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 + 5.53T + 31T^{2} \) |
| 37 | \( 1 - 8.71T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 - 8.63T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 + 3.95T + 79T^{2} \) |
| 83 | \( 1 + 5.25T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 + 2.67T + 97T^{2} \) |
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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
−8.164246710933097129709280390068, −7.66603828099486349846310102542, −7.14505903211928730934429142039, −6.04135898429135781697202585648, −5.27782528241680920210465513770, −4.26566389933729212294214050304, −3.88702462174909922337956909987, −2.71770284150304984995287320186, −2.14280048152680081369307610879, −1.61710263310366811897825590658,
1.61710263310366811897825590658, 2.14280048152680081369307610879, 2.71770284150304984995287320186, 3.88702462174909922337956909987, 4.26566389933729212294214050304, 5.27782528241680920210465513770, 6.04135898429135781697202585648, 7.14505903211928730934429142039, 7.66603828099486349846310102542, 8.164246710933097129709280390068
