| L(s) = 1 | − 2-s − 4-s − 0.561·5-s + 3·8-s + 0.561·10-s − 0.561·11-s + 0.561·13-s − 16-s + 3.12·19-s + 0.561·20-s + 0.561·22-s − 23-s − 4.68·25-s − 0.561·26-s + 29-s − 6.56·31-s − 5·32-s + 6.56·37-s − 3.12·38-s − 1.68·40-s − 3.43·41-s + 12.2·43-s + 0.561·44-s + 46-s + 1.12·47-s − 7·49-s + 4.68·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.5·4-s − 0.251·5-s + 1.06·8-s + 0.177·10-s − 0.169·11-s + 0.155·13-s − 0.250·16-s + 0.716·19-s + 0.125·20-s + 0.119·22-s − 0.208·23-s − 0.936·25-s − 0.110·26-s + 0.185·29-s − 1.17·31-s − 0.883·32-s + 1.07·37-s − 0.506·38-s − 0.266·40-s − 0.536·41-s + 1.86·43-s + 0.0846·44-s + 0.147·46-s + 0.163·47-s − 49-s + 0.662·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 0.561T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 - 7.36T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.69672575415617874989788090750, −7.45092790700056853720250896959, −6.33755197153873944713935072660, −5.57181464983287422614651152201, −4.79478620579140083835795130352, −4.05480843483099138363700848678, −3.30507888668318548641400306258, −2.11312584945081072998564455562, −1.10480956395666603084200917221, 0,
1.10480956395666603084200917221, 2.11312584945081072998564455562, 3.30507888668318548641400306258, 4.05480843483099138363700848678, 4.79478620579140083835795130352, 5.57181464983287422614651152201, 6.33755197153873944713935072660, 7.45092790700056853720250896959, 7.69672575415617874989788090750
