| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 7-s − 2.23·8-s + 0.236·11-s + 1.23·13-s + 0.618·14-s + 1.85·16-s − 2.47·17-s − 4.47·19-s + 0.145·22-s − 6.23·23-s + 0.763·26-s − 1.61·28-s − 5·29-s + 3.70·31-s + 5.61·32-s − 1.52·34-s + 3·37-s − 2.76·38-s − 4.76·41-s + 1.76·43-s − 0.381·44-s − 3.85·46-s + 2·47-s + 49-s − 2.00·52-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.377·7-s − 0.790·8-s + 0.0711·11-s + 0.342·13-s + 0.165·14-s + 0.463·16-s − 0.599·17-s − 1.02·19-s + 0.0311·22-s − 1.30·23-s + 0.149·26-s − 0.305·28-s − 0.928·29-s + 0.666·31-s + 0.993·32-s − 0.262·34-s + 0.493·37-s − 0.448·38-s − 0.744·41-s + 0.268·43-s − 0.0575·44-s − 0.568·46-s + 0.291·47-s + 0.142·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−8.936648158082366501215000495106, −8.355908125083163166920436300508, −7.54275012010090119320379393016, −6.29776151072495911938790822439, −5.78576543405928746189519590599, −4.59661848624300459361287539544, −4.19700420641347457049770580460, −3.07642602529909298248137749277, −1.74271250672807477164152803198, 0,
1.74271250672807477164152803198, 3.07642602529909298248137749277, 4.19700420641347457049770580460, 4.59661848624300459361287539544, 5.78576543405928746189519590599, 6.29776151072495911938790822439, 7.54275012010090119320379393016, 8.355908125083163166920436300508, 8.936648158082366501215000495106
