| L(s) = 1 | − 3·3-s + 4·5-s + 3·7-s + 6·9-s − 3·11-s + 7·13-s − 12·15-s − 2·17-s − 2·19-s − 9·21-s + 3·23-s + 11·25-s − 9·27-s + 8·29-s + 10·31-s + 9·33-s + 12·35-s + 8·37-s − 21·39-s − 6·41-s − 11·43-s + 24·45-s − 3·47-s + 2·49-s + 6·51-s + 4·53-s − 12·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s + 1.13·7-s + 2·9-s − 0.904·11-s + 1.94·13-s − 3.09·15-s − 0.485·17-s − 0.458·19-s − 1.96·21-s + 0.625·23-s + 11/5·25-s − 1.73·27-s + 1.48·29-s + 1.79·31-s + 1.56·33-s + 2.02·35-s + 1.31·37-s − 3.36·39-s − 0.937·41-s − 1.67·43-s + 3.57·45-s − 0.437·47-s + 2/7·49-s + 0.840·51-s + 0.549·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.495852930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.495852930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.23714474829372, −15.30989847111736, −15.17340902886045, −13.97520499726091, −13.62559968053784, −13.25540558274206, −12.68360977935102, −11.89131728329710, −11.35557374639384, −10.83699798463407, −10.45302787957781, −10.07790497835246, −9.247576375862251, −8.391420180605914, −8.098791892315150, −6.710471636212808, −6.533591914323978, −5.977368863402883, −5.418646868014121, −4.776997414085919, −4.532488753527222, −3.107411270759801, −2.123036815314907, −1.375346732390592, −0.8588697881619895,
0.8588697881619895, 1.375346732390592, 2.123036815314907, 3.107411270759801, 4.532488753527222, 4.776997414085919, 5.418646868014121, 5.977368863402883, 6.533591914323978, 6.710471636212808, 8.098791892315150, 8.391420180605914, 9.247576375862251, 10.07790497835246, 10.45302787957781, 10.83699798463407, 11.35557374639384, 11.89131728329710, 12.68360977935102, 13.25540558274206, 13.62559968053784, 13.97520499726091, 15.17340902886045, 15.30989847111736, 16.23714474829372
