| L(s) = 1 | + 3-s + 3·7-s + 9-s − 2·11-s + 13-s + 17-s + 2·19-s + 3·21-s + 4·23-s + 27-s + 5·29-s + 8·31-s − 2·33-s + 3·37-s + 39-s + 8·41-s + 13·43-s + 3·47-s + 2·49-s + 51-s + 2·57-s + 4·59-s − 14·61-s + 3·63-s − 2·67-s + 4·69-s + 14·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.242·17-s + 0.458·19-s + 0.654·21-s + 0.834·23-s + 0.192·27-s + 0.928·29-s + 1.43·31-s − 0.348·33-s + 0.493·37-s + 0.160·39-s + 1.24·41-s + 1.98·43-s + 0.437·47-s + 2/7·49-s + 0.140·51-s + 0.264·57-s + 0.520·59-s − 1.79·61-s + 0.377·63-s − 0.244·67-s + 0.481·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.880534798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.880534798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.14044626222983, −13.99715963159674, −13.07985940679295, −12.91768046949546, −12.11992270184986, −11.73130609282812, −11.11797253481803, −10.62097031372491, −10.28881252332039, −9.423788872856538, −9.161333440354021, −8.403831000277547, −8.069380418853670, −7.583598542833502, −7.147242915147807, −6.305510727112789, −5.792505737897175, −5.094769060778379, −4.565269729664079, −4.180146321309978, −3.245784288936994, −2.700686210548848, −2.203470986114635, −1.213044709139748, −0.8237284971467840,
0.8237284971467840, 1.213044709139748, 2.203470986114635, 2.700686210548848, 3.245784288936994, 4.180146321309978, 4.565269729664079, 5.094769060778379, 5.792505737897175, 6.305510727112789, 7.147242915147807, 7.583598542833502, 8.069380418853670, 8.403831000277547, 9.161333440354021, 9.423788872856538, 10.28881252332039, 10.62097031372491, 11.11797253481803, 11.73130609282812, 12.11992270184986, 12.91768046949546, 13.07985940679295, 13.99715963159674, 14.14044626222983
