| L(s) = 1 | − 1.18·2-s − 2.60·3-s − 6.60·4-s − 7.12·5-s + 3.07·6-s + 33.7·7-s + 17.2·8-s − 20.2·9-s + 8.41·10-s − 11·11-s + 17.2·12-s − 39.9·14-s + 18.5·15-s + 32.4·16-s + 36.4·17-s + 23.8·18-s − 97.0·19-s + 47.0·20-s − 87.9·21-s + 13.0·22-s + 31.1·23-s − 44.9·24-s − 74.2·25-s + 123.·27-s − 222.·28-s + 133.·29-s − 21.9·30-s + ⋯ |
| L(s) = 1 | − 0.417·2-s − 0.501·3-s − 0.825·4-s − 0.636·5-s + 0.209·6-s + 1.82·7-s + 0.762·8-s − 0.748·9-s + 0.266·10-s − 0.301·11-s + 0.413·12-s − 0.761·14-s + 0.319·15-s + 0.506·16-s + 0.520·17-s + 0.312·18-s − 1.17·19-s + 0.525·20-s − 0.914·21-s + 0.125·22-s + 0.282·23-s − 0.382·24-s − 0.594·25-s + 0.876·27-s − 1.50·28-s + 0.855·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.18T + 8T^{2} \) |
| 3 | \( 1 + 2.60T + 27T^{2} \) |
| 5 | \( 1 + 7.12T + 125T^{2} \) |
| 7 | \( 1 - 33.7T + 343T^{2} \) |
| 17 | \( 1 - 36.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 130.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 22.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 416.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 30.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 0.0701T + 1.48e5T^{2} \) |
| 59 | \( 1 - 27.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 609.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 895.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 215.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 784.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 91.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 818.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 49.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 186.T + 9.12e5T^{2} \) |
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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
−8.346889485148296709949467037409, −8.037696742280534663396056717527, −7.18139753862121841936723250500, −5.86159913354063143592768793059, −5.09019286917538570854600918850, −4.57207520439382032329894483014, −3.64909977508743091879638366244, −2.14098717711136012004566002669, −1.00111250433782951187647966530, 0,
1.00111250433782951187647966530, 2.14098717711136012004566002669, 3.64909977508743091879638366244, 4.57207520439382032329894483014, 5.09019286917538570854600918850, 5.86159913354063143592768793059, 7.18139753862121841936723250500, 8.037696742280534663396056717527, 8.346889485148296709949467037409
