| L(s) = 1 | + 3-s − 3.63·5-s + 4.36·7-s + 9-s − 2.73·11-s − 6.14·13-s − 3.63·15-s + 1.31·17-s + 5.38·19-s + 4.36·21-s + 8.22·25-s + 27-s − 6.53·29-s + 4.09·31-s − 2.73·33-s − 15.8·35-s − 5.76·37-s − 6.14·39-s + 7.77·41-s + 9.41·43-s − 3.63·45-s − 5.07·47-s + 12.0·49-s + 1.31·51-s − 4.08·53-s + 9.94·55-s + 5.38·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.62·5-s + 1.65·7-s + 0.333·9-s − 0.824·11-s − 1.70·13-s − 0.938·15-s + 0.319·17-s + 1.23·19-s + 0.953·21-s + 1.64·25-s + 0.192·27-s − 1.21·29-s + 0.734·31-s − 0.476·33-s − 2.68·35-s − 0.948·37-s − 0.984·39-s + 1.21·41-s + 1.43·43-s − 0.542·45-s − 0.740·47-s + 1.72·49-s + 0.184·51-s − 0.560·53-s + 1.34·55-s + 0.713·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.787065417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787065417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 29 | \( 1 + 6.53T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 - 7.77T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 - 4.40T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 - 7.19T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 + 2.44T + 97T^{2} \) |
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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
−7.79458626006149282398276490872, −7.53326202858854846806067374804, −7.25254825172035403748182647548, −5.66484020806324908992288773314, −4.81104453098235147899134480158, −4.61399806032373959945354675197, −3.60956809484106378591617822268, −2.81829805888796013502464472001, −1.93398507043432496766137069638, −0.66932091577241356965660580020,
0.66932091577241356965660580020, 1.93398507043432496766137069638, 2.81829805888796013502464472001, 3.60956809484106378591617822268, 4.61399806032373959945354675197, 4.81104453098235147899134480158, 5.66484020806324908992288773314, 7.25254825172035403748182647548, 7.53326202858854846806067374804, 7.79458626006149282398276490872
