| L(s) = 1 | + 0.601·3-s + 4.19·5-s − 7-s − 2.63·9-s + 2.71·11-s − 5.83·13-s + 2.52·15-s − 17-s − 6.47·19-s − 0.601·21-s + 3.86·23-s + 12.6·25-s − 3.39·27-s − 0.714·29-s − 9.96·31-s + 1.63·33-s − 4.19·35-s − 7.80·37-s − 3.51·39-s − 3.82·41-s − 0.681·43-s − 11.0·45-s + 7.53·47-s + 49-s − 0.601·51-s + 7.25·53-s + 11.4·55-s + ⋯ |
| L(s) = 1 | + 0.347·3-s + 1.87·5-s − 0.377·7-s − 0.879·9-s + 0.818·11-s − 1.61·13-s + 0.652·15-s − 0.242·17-s − 1.48·19-s − 0.131·21-s + 0.806·23-s + 2.52·25-s − 0.652·27-s − 0.132·29-s − 1.79·31-s + 0.284·33-s − 0.709·35-s − 1.28·37-s − 0.562·39-s − 0.597·41-s − 0.103·43-s − 1.65·45-s + 1.09·47-s + 0.142·49-s − 0.0842·51-s + 0.996·53-s + 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.601T + 3T^{2} \) |
| 5 | \( 1 - 4.19T + 5T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 + 5.83T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 3.86T + 23T^{2} \) |
| 29 | \( 1 + 0.714T + 29T^{2} \) |
| 31 | \( 1 + 9.96T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 + 0.681T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 - 7.25T + 53T^{2} \) |
| 59 | \( 1 - 1.29T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 6.76T + 71T^{2} \) |
| 73 | \( 1 - 1.60T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 8.58T + 83T^{2} \) |
| 89 | \( 1 + 9.67T + 89T^{2} \) |
| 97 | \( 1 - 9.18T + 97T^{2} \) |
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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
−7.20480150108973674110840578132, −6.90730132317112391872175228436, −6.04356036531046363051993805422, −5.53421407790956856626629136508, −4.90146310945538882955648429642, −3.85922343394869849497238361758, −2.80889152618158415774380077422, −2.29737202665912778932448437474, −1.60035557518999633928839548955, 0,
1.60035557518999633928839548955, 2.29737202665912778932448437474, 2.80889152618158415774380077422, 3.85922343394869849497238361758, 4.90146310945538882955648429642, 5.53421407790956856626629136508, 6.04356036531046363051993805422, 6.90730132317112391872175228436, 7.20480150108973674110840578132
