| L(s) = 1 | − 7.32·3-s + 1.19e3·5-s − 2.40e3·7-s − 1.96e4·9-s − 1.82e4·11-s + 5.03e4·13-s − 8.72e3·15-s + 2.83e5·17-s − 2.73e5·19-s + 1.75e4·21-s − 1.06e6·23-s − 5.34e5·25-s + 2.87e5·27-s − 5.40e6·29-s − 1.75e6·31-s + 1.33e5·33-s − 2.85e6·35-s − 7.76e6·37-s − 3.68e5·39-s + 8.92e6·41-s − 3.46e7·43-s − 2.33e7·45-s − 3.66e7·47-s + 5.76e6·49-s − 2.07e6·51-s − 6.10e7·53-s − 2.17e7·55-s + ⋯ |
| L(s) = 1 | − 0.0521·3-s + 0.852·5-s − 0.377·7-s − 0.997·9-s − 0.375·11-s + 0.488·13-s − 0.0444·15-s + 0.823·17-s − 0.481·19-s + 0.0197·21-s − 0.790·23-s − 0.273·25-s + 0.104·27-s − 1.41·29-s − 0.342·31-s + 0.0196·33-s − 0.322·35-s − 0.681·37-s − 0.0255·39-s + 0.493·41-s − 1.54·43-s − 0.850·45-s − 1.09·47-s + 0.142·49-s − 0.0429·51-s − 1.06·53-s − 0.320·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 + 7.32T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.19e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 1.82e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.03e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.83e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.73e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.06e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.76e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 8.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.46e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.66e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.02e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.52e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.18e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.33e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.06e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.89e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.73e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.97e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.82e8T + 7.60e17T^{2} \) |
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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
−12.90367479226140293384968477520, −11.57582344976508275654099941679, −10.35141971554670286467114754075, −9.279480998333914652431345306713, −8.001767240894191431153840589958, −6.28982032879519245680865611148, −5.39876850179180110601858479412, −3.40188700693229396626999332687, −1.90583267238378760935328241160, 0,
1.90583267238378760935328241160, 3.40188700693229396626999332687, 5.39876850179180110601858479412, 6.28982032879519245680865611148, 8.001767240894191431153840589958, 9.279480998333914652431345306713, 10.35141971554670286467114754075, 11.57582344976508275654099941679, 12.90367479226140293384968477520
