| L(s) = 1 | − 1.65·3-s − 2.91·5-s − 0.255·9-s + 11-s − 0.343·13-s + 4.82·15-s + 1.25·17-s + 4.39·19-s + 8.70·23-s + 3.48·25-s + 5.39·27-s + 5.22·29-s − 9.73·31-s − 1.65·33-s + 2.17·37-s + 0.568·39-s − 9.99·41-s − 3.79·43-s + 0.744·45-s − 9.70·47-s − 2.08·51-s − 4.34·53-s − 2.91·55-s − 7.27·57-s + 4.99·59-s + 0.511·61-s + 65-s + ⋯ |
| L(s) = 1 | − 0.956·3-s − 1.30·5-s − 0.0852·9-s + 0.301·11-s − 0.0952·13-s + 1.24·15-s + 0.304·17-s + 1.00·19-s + 1.81·23-s + 0.696·25-s + 1.03·27-s + 0.970·29-s − 1.74·31-s − 0.288·33-s + 0.357·37-s + 0.0910·39-s − 1.56·41-s − 0.578·43-s + 0.110·45-s − 1.41·47-s − 0.291·51-s − 0.596·53-s − 0.392·55-s − 0.964·57-s + 0.649·59-s + 0.0654·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 + 2.91T + 5T^{2} \) |
| 13 | \( 1 + 0.343T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 - 4.39T + 19T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 + 9.73T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 9.99T + 41T^{2} \) |
| 43 | \( 1 + 3.79T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 - 4.99T + 59T^{2} \) |
| 61 | \( 1 - 0.511T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 73 | \( 1 - 7.28T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 + 1.14T + 97T^{2} \) |
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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
−8.614711496269650076393770884659, −7.84571927772915964006822975864, −7.06156892111758091911946536898, −6.46144291448875524985003605157, −5.25285577588910344806110232343, −4.91676274859652781926695113938, −3.69916054482749458103697846082, −3.02668771153405704820279894340, −1.19802951458921137436160928554, 0,
1.19802951458921137436160928554, 3.02668771153405704820279894340, 3.69916054482749458103697846082, 4.91676274859652781926695113938, 5.25285577588910344806110232343, 6.46144291448875524985003605157, 7.06156892111758091911946536898, 7.84571927772915964006822975864, 8.614711496269650076393770884659
