| L(s) = 1 | − 1.95·2-s + 3-s + 1.82·4-s − 1.95·6-s + 1.54·7-s + 0.338·8-s + 9-s + 1.82·12-s + 4.15·13-s − 3.02·14-s − 4.31·16-s + 0.741·17-s − 1.95·18-s − 7.48·19-s + 1.54·21-s + 1.15·23-s + 0.338·24-s − 8.13·26-s + 27-s + 2.82·28-s − 6.88·29-s + 6.77·31-s + 7.76·32-s − 1.45·34-s + 1.82·36-s − 3.92·37-s + 14.6·38-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.577·3-s + 0.913·4-s − 0.798·6-s + 0.584·7-s + 0.119·8-s + 0.333·9-s + 0.527·12-s + 1.15·13-s − 0.809·14-s − 1.07·16-s + 0.179·17-s − 0.461·18-s − 1.71·19-s + 0.337·21-s + 0.241·23-s + 0.0690·24-s − 1.59·26-s + 0.192·27-s + 0.534·28-s − 1.27·29-s + 1.21·31-s + 1.37·32-s − 0.248·34-s + 0.304·36-s − 0.644·37-s + 2.37·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 - 0.741T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 + 6.88T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 + 3.92T + 37T^{2} \) |
| 41 | \( 1 + 0.0345T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 7.04T + 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 - 3.55T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 + 6.17T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 0.144T + 97T^{2} \) |
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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
−7.76105683606242510725959397551, −6.86714393272579403735870498526, −6.39407812751991463963804389520, −5.36836886445807676523364208375, −4.43099268772835460908141286315, −3.84660818589328531445829874461, −2.76706745227759346555886894179, −1.82784734026728706459808724839, −1.30850266896252076702205213789, 0,
1.30850266896252076702205213789, 1.82784734026728706459808724839, 2.76706745227759346555886894179, 3.84660818589328531445829874461, 4.43099268772835460908141286315, 5.36836886445807676523364208375, 6.39407812751991463963804389520, 6.86714393272579403735870498526, 7.76105683606242510725959397551
