| L(s) = 1 | + 0.688·2-s − 1.52·4-s − 4.90·7-s − 2.42·8-s − 3.21·11-s + 0.836·13-s − 3.37·14-s + 1.37·16-s + 6.73·17-s + 3·19-s − 2.21·22-s + 5.21·23-s + 0.576·26-s + 7.47·28-s + 1.49·29-s + 31-s + 5.80·32-s + 4.64·34-s − 0.407·37-s + 2.06·38-s − 9.54·41-s + 7.61·43-s + 4.90·44-s + 3.59·46-s − 6.62·47-s + 17.0·49-s − 1.27·52-s + ⋯ |
| L(s) = 1 | + 0.487·2-s − 0.762·4-s − 1.85·7-s − 0.858·8-s − 0.969·11-s + 0.232·13-s − 0.902·14-s + 0.344·16-s + 1.63·17-s + 0.688·19-s − 0.472·22-s + 1.08·23-s + 0.113·26-s + 1.41·28-s + 0.277·29-s + 0.179·31-s + 1.02·32-s + 0.796·34-s − 0.0670·37-s + 0.335·38-s − 1.49·41-s + 1.16·43-s + 0.739·44-s + 0.529·46-s − 0.965·47-s + 2.43·49-s − 0.176·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.688T + 2T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 0.836T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 37 | \( 1 + 0.407T + 37T^{2} \) |
| 41 | \( 1 + 9.54T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 61 | \( 1 - 0.755T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 - 8.29T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 2.78T + 89T^{2} \) |
| 97 | \( 1 + 8.95T + 97T^{2} \) |
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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
−7.58217387195023104055332521475, −6.77417623031062437807610965152, −6.06115027883150622638499407881, −5.43214016307418270573119207609, −4.89499783242131668669954485326, −3.76364131549396822913844997766, −3.20712237580641484667379486717, −2.81407668311107737005337219282, −1.03572161951298536661248239314, 0,
1.03572161951298536661248239314, 2.81407668311107737005337219282, 3.20712237580641484667379486717, 3.76364131549396822913844997766, 4.89499783242131668669954485326, 5.43214016307418270573119207609, 6.06115027883150622638499407881, 6.77417623031062437807610965152, 7.58217387195023104055332521475
