| L(s) = 1 | − 2.17·2-s + 1.41·3-s + 2.70·4-s − 3.06·6-s + 1.76·7-s − 1.53·8-s − 1.00·9-s + 1.83·11-s + 3.82·12-s + 2.60·13-s − 3.82·14-s − 2.07·16-s − 4.23·17-s + 2.17·18-s + 2.48·21-s − 3.98·22-s + 2.20·23-s − 2.17·24-s − 5.65·26-s − 5.65·27-s + 4.77·28-s − 7.12·29-s + 0.303·31-s + 7.58·32-s + 2.59·33-s + 9.19·34-s − 2.72·36-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 0.815·3-s + 1.35·4-s − 1.25·6-s + 0.665·7-s − 0.544·8-s − 0.334·9-s + 0.554·11-s + 1.10·12-s + 0.722·13-s − 1.02·14-s − 0.519·16-s − 1.02·17-s + 0.513·18-s + 0.543·21-s − 0.850·22-s + 0.459·23-s − 0.443·24-s − 1.10·26-s − 1.08·27-s + 0.902·28-s − 1.32·29-s + 0.0545·31-s + 1.34·32-s + 0.452·33-s + 1.57·34-s − 0.453·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 1.76T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 23 | \( 1 - 2.20T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 0.303T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 - 7.88T + 79T^{2} \) |
| 83 | \( 1 + 6.93T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.75T + 97T^{2} \) |
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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
−7.76146004534449371008594931157, −6.99577866705735556029100078398, −6.35048419859340212909899857638, −5.45903684032668128762472405691, −4.44698210388984906393047338145, −3.70264609540462482961027922819, −2.71381159475707967836875432660, −1.93853977782619403454249606637, −1.27766268809669061371292809971, 0,
1.27766268809669061371292809971, 1.93853977782619403454249606637, 2.71381159475707967836875432660, 3.70264609540462482961027922819, 4.44698210388984906393047338145, 5.45903684032668128762472405691, 6.35048419859340212909899857638, 6.99577866705735556029100078398, 7.76146004534449371008594931157
