| L(s) = 1 | − 2.74·2-s + 3-s + 5.52·4-s + 0.811·5-s − 2.74·6-s − 9.68·8-s + 9-s − 2.22·10-s − 4.34·11-s + 5.52·12-s + 2.30·13-s + 0.811·15-s + 15.5·16-s − 0.928·17-s − 2.74·18-s − 4.60·19-s + 4.48·20-s + 11.9·22-s + 0.827·23-s − 9.68·24-s − 4.34·25-s − 6.31·26-s + 27-s + 3.78·29-s − 2.22·30-s + 5.65·31-s − 23.2·32-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.76·4-s + 0.362·5-s − 1.12·6-s − 3.42·8-s + 0.333·9-s − 0.704·10-s − 1.30·11-s + 1.59·12-s + 0.638·13-s + 0.209·15-s + 3.87·16-s − 0.225·17-s − 0.646·18-s − 1.05·19-s + 1.00·20-s + 2.53·22-s + 0.172·23-s − 1.97·24-s − 0.868·25-s − 1.23·26-s + 0.192·27-s + 0.702·29-s − 0.406·30-s + 1.01·31-s − 4.10·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 - 0.811T + 5T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 0.928T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 23 | \( 1 - 0.827T + 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 1.48T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 2.35T + 61T^{2} \) |
| 67 | \( 1 - 8.74T + 67T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 - 4.25T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.964690600794652829984761260096, −7.40275819205561027514385458212, −6.47803397836064653977904495032, −6.07928757725862865476519934156, −4.97671706799637354831098402028, −3.66377107128757254063936895325, −2.64218604165871862023141742346, −2.21749598562916639504175686277, −1.21733912357512425717582194837, 0,
1.21733912357512425717582194837, 2.21749598562916639504175686277, 2.64218604165871862023141742346, 3.66377107128757254063936895325, 4.97671706799637354831098402028, 6.07928757725862865476519934156, 6.47803397836064653977904495032, 7.40275819205561027514385458212, 7.964690600794652829984761260096
