| L(s) = 1 | + 1.82·2-s + 1.32·4-s − 1.45·7-s − 1.23·8-s + 3.89·11-s − 3.05·13-s − 2.64·14-s − 4.89·16-s + 3.92·17-s − 19-s + 7.10·22-s − 5.37·23-s − 5.57·26-s − 1.91·28-s − 6·29-s − 8.43·31-s − 6.45·32-s + 7.14·34-s + 5.95·37-s − 1.82·38-s − 10.4·41-s + 1.45·43-s + 5.14·44-s − 9.79·46-s + 4.90·47-s − 4.89·49-s − 4.04·52-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.660·4-s − 0.548·7-s − 0.437·8-s + 1.17·11-s − 0.848·13-s − 0.706·14-s − 1.22·16-s + 0.951·17-s − 0.229·19-s + 1.51·22-s − 1.12·23-s − 1.09·26-s − 0.362·28-s − 1.11·29-s − 1.51·31-s − 1.14·32-s + 1.22·34-s + 0.979·37-s − 0.295·38-s − 1.62·41-s + 0.221·43-s + 0.776·44-s − 1.44·46-s + 0.715·47-s − 0.699·49-s − 0.560·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 3.89T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 + 4.23T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 9.84T + 67T^{2} \) |
| 71 | \( 1 + 8.64T + 71T^{2} \) |
| 73 | \( 1 - 2.43T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.05T + 97T^{2} \) |
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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
−7.84049193134921126295278986020, −7.02873717441575651650930965587, −6.38253125514339303196966312465, −5.67935020080422081429017308439, −5.08650166255175410218448848259, −4.00046074058463134947503735020, −3.72496894954156675055169882685, −2.75650619466820822471354067945, −1.71060540388493841649297305056, 0,
1.71060540388493841649297305056, 2.75650619466820822471354067945, 3.72496894954156675055169882685, 4.00046074058463134947503735020, 5.08650166255175410218448848259, 5.67935020080422081429017308439, 6.38253125514339303196966312465, 7.02873717441575651650930965587, 7.84049193134921126295278986020
