| L(s) = 1 | − 3-s − 3·7-s + 9-s + 3·11-s + 13-s + 3·17-s − 2·19-s + 3·21-s + 23-s − 27-s + 8·29-s + 4·31-s − 3·33-s − 5·37-s − 39-s − 7·41-s + 2·43-s + 4·47-s + 2·49-s − 3·51-s − 11·53-s + 2·57-s + 4·59-s − 61-s − 3·63-s − 69-s + 5·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.654·21-s + 0.208·23-s − 0.192·27-s + 1.48·29-s + 0.718·31-s − 0.522·33-s − 0.821·37-s − 0.160·39-s − 1.09·41-s + 0.304·43-s + 0.583·47-s + 2/7·49-s − 0.420·51-s − 1.51·53-s + 0.264·57-s + 0.520·59-s − 0.128·61-s − 0.377·63-s − 0.120·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.649549439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649549439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.26965134516852, −13.70869909901364, −13.26327226138480, −12.51653512076678, −12.38876226941536, −11.77666519808668, −11.34723261736424, −10.58233356354525, −10.21639168943354, −9.772026863392263, −9.204936485829937, −8.649045611736202, −8.128807743064077, −7.356930418420853, −6.741825182661449, −6.411058312155213, −6.029862108013176, −5.259078749164333, −4.677307995624336, −4.034427198152939, −3.373384495351351, −2.967393510559684, −1.981085876928486, −1.175497282807092, −0.5046607972339255,
0.5046607972339255, 1.175497282807092, 1.981085876928486, 2.967393510559684, 3.373384495351351, 4.034427198152939, 4.677307995624336, 5.259078749164333, 6.029862108013176, 6.411058312155213, 6.741825182661449, 7.356930418420853, 8.128807743064077, 8.649045611736202, 9.204936485829937, 9.772026863392263, 10.21639168943354, 10.58233356354525, 11.34723261736424, 11.77666519808668, 12.38876226941536, 12.51653512076678, 13.26327226138480, 13.70869909901364, 14.26965134516852
