| L(s) = 1 | + 0.378·2-s − 3-s − 1.85·4-s − 5-s − 0.378·6-s + 4.99·7-s − 1.45·8-s + 9-s − 0.378·10-s − 5.16·11-s + 1.85·12-s − 0.273·13-s + 1.89·14-s + 15-s + 3.16·16-s + 1.83·17-s + 0.378·18-s + 4.45·19-s + 1.85·20-s − 4.99·21-s − 1.95·22-s − 6.85·23-s + 1.45·24-s + 25-s − 0.103·26-s − 27-s − 9.28·28-s + ⋯ |
| L(s) = 1 | + 0.267·2-s − 0.577·3-s − 0.928·4-s − 0.447·5-s − 0.154·6-s + 1.88·7-s − 0.515·8-s + 0.333·9-s − 0.119·10-s − 1.55·11-s + 0.536·12-s − 0.0758·13-s + 0.505·14-s + 0.258·15-s + 0.790·16-s + 0.445·17-s + 0.0891·18-s + 1.02·19-s + 0.415·20-s − 1.09·21-s − 0.416·22-s − 1.43·23-s + 0.297·24-s + 0.200·25-s − 0.0202·26-s − 0.192·27-s − 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.378T + 2T^{2} \) |
| 7 | \( 1 - 4.99T + 7T^{2} \) |
| 11 | \( 1 + 5.16T + 11T^{2} \) |
| 13 | \( 1 + 0.273T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 - 4.45T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 - 0.384T + 29T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 - 6.08T + 53T^{2} \) |
| 59 | \( 1 + 7.53T + 59T^{2} \) |
| 61 | \( 1 + 5.36T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 8.37T + 71T^{2} \) |
| 73 | \( 1 + 4.58T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 + 0.154T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.67467800541735031725302636955, −7.42121905925969609447950892572, −5.86791359916926465064360134039, −5.40115323943952644521345052351, −4.91436374520680219276479092592, −4.30236819488106997642618734704, −3.46035193619079804131777714218, −2.26236501920272328725800686957, −1.16932805914188024273046909348, 0,
1.16932805914188024273046909348, 2.26236501920272328725800686957, 3.46035193619079804131777714218, 4.30236819488106997642618734704, 4.91436374520680219276479092592, 5.40115323943952644521345052351, 5.86791359916926465064360134039, 7.42121905925969609447950892572, 7.67467800541735031725302636955
