L(s) = 1 | − 2.76·3-s + 1.62·5-s + 4.62·9-s − 4.49·11-s − 0.103·13-s − 4.49·15-s + 2·17-s + 7.25·19-s − 5.52·23-s − 2.35·25-s − 4.49·27-s − 29-s − 6.76·31-s + 12.4·33-s − 5.25·37-s + 0.284·39-s + 5.79·41-s + 10.0·43-s + 7.52·45-s + 11.5·47-s − 7·49-s − 5.52·51-s + 7.14·53-s − 7.30·55-s − 20.0·57-s + 1.52·59-s − 9.04·61-s + ⋯ |
L(s) = 1 | − 1.59·3-s + 0.727·5-s + 1.54·9-s − 1.35·11-s − 0.0285·13-s − 1.15·15-s + 0.485·17-s + 1.66·19-s − 1.15·23-s − 0.471·25-s − 0.864·27-s − 0.185·29-s − 1.21·31-s + 2.15·33-s − 0.863·37-s + 0.0455·39-s + 0.904·41-s + 1.52·43-s + 1.12·45-s + 1.68·47-s − 49-s − 0.773·51-s + 0.982·53-s − 0.984·55-s − 2.65·57-s + 0.198·59-s − 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 + 0.103T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 31 | \( 1 + 6.76T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−9.094841189483888813449118831710, −7.66274759331615056564596168996, −7.35415163373785081226255233041, −6.04997363806133720166089019663, −5.63451580798356693331713996935, −5.18959555993701630854912359154, −4.05279742046282290633945593517, −2.68064428940024873700939033743, −1.39338316208286742877005505906, 0,
1.39338316208286742877005505906, 2.68064428940024873700939033743, 4.05279742046282290633945593517, 5.18959555993701630854912359154, 5.63451580798356693331713996935, 6.04997363806133720166089019663, 7.35415163373785081226255233041, 7.66274759331615056564596168996, 9.094841189483888813449118831710