| L(s) = 1 | − 1.87·3-s − 3.24·5-s − 1.92·7-s + 0.507·9-s + 11-s + 2.85·13-s + 6.08·15-s − 2.33·17-s − 19-s + 3.60·21-s + 2.74·23-s + 5.54·25-s + 4.66·27-s − 0.972·29-s + 0.00551·31-s − 1.87·33-s + 6.24·35-s + 9.67·37-s − 5.33·39-s + 6.65·41-s − 7.99·43-s − 1.64·45-s − 3.46·47-s − 3.30·49-s + 4.36·51-s + 10.5·53-s − 3.24·55-s + ⋯ |
| L(s) = 1 | − 1.08·3-s − 1.45·5-s − 0.726·7-s + 0.169·9-s + 0.301·11-s + 0.790·13-s + 1.57·15-s − 0.565·17-s − 0.229·19-s + 0.786·21-s + 0.572·23-s + 1.10·25-s + 0.898·27-s − 0.180·29-s + 0.000989·31-s − 0.326·33-s + 1.05·35-s + 1.58·37-s − 0.855·39-s + 1.03·41-s − 1.21·43-s − 0.245·45-s − 0.506·47-s − 0.471·49-s + 0.611·51-s + 1.44·53-s − 0.437·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 2.33T + 17T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 0.972T + 29T^{2} \) |
| 31 | \( 1 - 0.00551T + 31T^{2} \) |
| 37 | \( 1 - 9.67T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 + 7.99T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 1.87T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 7.57T + 97T^{2} \) |
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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
−8.286407808377881572725545695126, −7.37383328551294933699164039564, −6.65325016611048000503213465489, −6.12946931217157435658562202734, −5.21166743424298955650863640625, −4.29874566428575760219475594647, −3.71425661888707943068421178394, −2.73816446387778668869635994370, −0.977762279250492598084285917564, 0,
0.977762279250492598084285917564, 2.73816446387778668869635994370, 3.71425661888707943068421178394, 4.29874566428575760219475594647, 5.21166743424298955650863640625, 6.12946931217157435658562202734, 6.65325016611048000503213465489, 7.37383328551294933699164039564, 8.286407808377881572725545695126
