| L(s) = 1 | − 3.73·5-s − 6.46·13-s + 5.73·17-s + 8.92·25-s + 10.6·29-s + 9.39·37-s + 8·41-s − 7·49-s − 4·53-s − 15.3·61-s + 24.1·65-s − 16.8·73-s − 21.3·85-s − 0.660·89-s − 18·97-s − 20·101-s + 14.3·109-s − 4.12·113-s + ⋯ |
| L(s) = 1 | − 1.66·5-s − 1.79·13-s + 1.39·17-s + 1.78·25-s + 1.97·29-s + 1.54·37-s + 1.24·41-s − 49-s − 0.549·53-s − 1.97·61-s + 2.99·65-s − 1.97·73-s − 2.32·85-s − 0.0699·89-s − 1.82·97-s − 1.99·101-s + 1.37·109-s − 0.387·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 0.660T + 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−7.80943779052706560209886633236, −7.38924459704379828540474071132, −6.59394465722425304341457255592, −5.59410678323158743238873034474, −4.58346329669266778116916996326, −4.38027692872652530015933425313, −3.16379874924908033649163286040, −2.72355090716680571135290985428, −1.09925122458770635824964610866, 0,
1.09925122458770635824964610866, 2.72355090716680571135290985428, 3.16379874924908033649163286040, 4.38027692872652530015933425313, 4.58346329669266778116916996326, 5.59410678323158743238873034474, 6.59394465722425304341457255592, 7.38924459704379828540474071132, 7.80943779052706560209886633236
