| L(s) = 1 | − 2.58·2-s + 3.02·3-s + 4.66·4-s − 5-s − 7.82·6-s + 4.31·7-s − 6.87·8-s + 6.18·9-s + 2.58·10-s + 11-s + 14.1·12-s − 7.03·13-s − 11.1·14-s − 3.02·15-s + 8.41·16-s − 6.76·17-s − 15.9·18-s − 8.08·19-s − 4.66·20-s + 13.0·21-s − 2.58·22-s − 6.49·23-s − 20.8·24-s + 25-s + 18.1·26-s + 9.63·27-s + 20.1·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 1.74·3-s + 2.33·4-s − 0.447·5-s − 3.19·6-s + 1.63·7-s − 2.43·8-s + 2.06·9-s + 0.816·10-s + 0.301·11-s + 4.07·12-s − 1.95·13-s − 2.97·14-s − 0.782·15-s + 2.10·16-s − 1.64·17-s − 3.76·18-s − 1.85·19-s − 1.04·20-s + 2.85·21-s − 0.550·22-s − 1.35·23-s − 4.25·24-s + 0.200·25-s + 3.56·26-s + 1.85·27-s + 3.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 - 3.02T + 3T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 13 | \( 1 + 7.03T + 13T^{2} \) |
| 17 | \( 1 + 6.76T + 17T^{2} \) |
| 19 | \( 1 + 8.08T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 3.00T + 37T^{2} \) |
| 41 | \( 1 + 8.12T + 41T^{2} \) |
| 43 | \( 1 + 1.69T + 43T^{2} \) |
| 47 | \( 1 - 1.27T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 + 0.966T + 67T^{2} \) |
| 71 | \( 1 + 9.01T + 71T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 5.64T + 97T^{2} \) |
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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
−8.163471037169744487590475664059, −7.892118667881393724827102953193, −7.08486513513193984499382246096, −6.54049469710527417054264921259, −4.68677778727689331871712550099, −4.26745707840215285647590278007, −2.79561829572780236948717857115, −2.05774779013176752901965104926, −1.78707213977423456258096654809, 0,
1.78707213977423456258096654809, 2.05774779013176752901965104926, 2.79561829572780236948717857115, 4.26745707840215285647590278007, 4.68677778727689331871712550099, 6.54049469710527417054264921259, 7.08486513513193984499382246096, 7.892118667881393724827102953193, 8.163471037169744487590475664059
