| L(s) = 1 | − 2.57·3-s − 0.393·5-s − 1.17·7-s + 3.61·9-s + 3.41·11-s + 6.91·13-s + 1.01·15-s − 17-s + 7.90·19-s + 3.01·21-s + 0.587·23-s − 4.84·25-s − 1.58·27-s + 1.81·29-s − 0.548·31-s − 8.77·33-s + 0.462·35-s + 1.80·37-s − 17.7·39-s − 3.26·41-s + 4.28·43-s − 1.42·45-s + 3.73·47-s − 5.62·49-s + 2.57·51-s − 2.18·53-s − 1.34·55-s + ⋯ |
| L(s) = 1 | − 1.48·3-s − 0.176·5-s − 0.443·7-s + 1.20·9-s + 1.02·11-s + 1.91·13-s + 0.261·15-s − 0.242·17-s + 1.81·19-s + 0.658·21-s + 0.122·23-s − 0.969·25-s − 0.304·27-s + 0.337·29-s − 0.0984·31-s − 1.52·33-s + 0.0781·35-s + 0.296·37-s − 2.84·39-s − 0.510·41-s + 0.653·43-s − 0.212·45-s + 0.545·47-s − 0.803·49-s + 0.360·51-s − 0.300·53-s − 0.181·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.205523584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.205523584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.57T + 3T^{2} \) |
| 5 | \( 1 + 0.393T + 5T^{2} \) |
| 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 6.91T + 13T^{2} \) |
| 19 | \( 1 - 7.90T + 19T^{2} \) |
| 23 | \( 1 - 0.587T + 23T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 + 0.548T + 31T^{2} \) |
| 37 | \( 1 - 1.80T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 + 2.18T + 53T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 8.79T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 6.24T + 89T^{2} \) |
| 97 | \( 1 - 1.99T + 97T^{2} \) |
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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
−8.479032805250239455524996170229, −7.52906961131070939127499784491, −6.73049451906495225198152533831, −6.12167410073524108598833678013, −5.73145198414507488908749224116, −4.78130579284465029898581285969, −3.89174740278658485331538623233, −3.24247124271739038808273447477, −1.50813504038994219856843287954, −0.75097642137035221348323705499,
0.75097642137035221348323705499, 1.50813504038994219856843287954, 3.24247124271739038808273447477, 3.89174740278658485331538623233, 4.78130579284465029898581285969, 5.73145198414507488908749224116, 6.12167410073524108598833678013, 6.73049451906495225198152533831, 7.52906961131070939127499784491, 8.479032805250239455524996170229
