| L(s) = 1 | − 0.383·3-s − 4.10·5-s + 0.287·7-s − 2.85·9-s + 0.513·11-s + 2.10·13-s + 1.57·15-s − 1.15·17-s − 19-s − 0.110·21-s + 8.21·23-s + 11.8·25-s + 2.24·27-s − 0.842·29-s + 5.84·31-s − 0.197·33-s − 1.17·35-s − 4.24·37-s − 0.809·39-s + 3.80·41-s − 1.27·43-s + 11.7·45-s + 5.24·47-s − 6.91·49-s + 0.443·51-s + 53-s − 2.10·55-s + ⋯ |
| L(s) = 1 | − 0.221·3-s − 1.83·5-s + 0.108·7-s − 0.950·9-s + 0.154·11-s + 0.584·13-s + 0.406·15-s − 0.279·17-s − 0.229·19-s − 0.0240·21-s + 1.71·23-s + 2.36·25-s + 0.432·27-s − 0.156·29-s + 1.04·31-s − 0.0343·33-s − 0.199·35-s − 0.698·37-s − 0.129·39-s + 0.594·41-s − 0.194·43-s + 1.74·45-s + 0.765·47-s − 0.988·49-s + 0.0620·51-s + 0.137·53-s − 0.284·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 0.383T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 - 0.287T + 7T^{2} \) |
| 11 | \( 1 - 0.513T + 11T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 + 0.842T + 29T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 3.80T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 59 | \( 1 - 3.73T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 0.286T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 + 7.34T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 9.32T + 83T^{2} \) |
| 89 | \( 1 - 2.32T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.207042397931381800037566640185, −7.34502426704659470720896681100, −6.76998659543215347754493846332, −5.87169473017293860150098771160, −4.89175091894905509832236871469, −4.30013340416860122225690225917, −3.38720698447819154882616364444, −2.80280424569598013711284870959, −1.09639042566798634078494325829, 0,
1.09639042566798634078494325829, 2.80280424569598013711284870959, 3.38720698447819154882616364444, 4.30013340416860122225690225917, 4.89175091894905509832236871469, 5.87169473017293860150098771160, 6.76998659543215347754493846332, 7.34502426704659470720896681100, 8.207042397931381800037566640185
