| L(s) = 1 | − 3-s − 4·7-s + 9-s − 2·11-s − 13-s + 6·17-s + 4·19-s + 4·21-s + 4·23-s − 27-s + 6·29-s − 8·31-s + 2·33-s − 10·37-s + 39-s − 4·41-s + 4·43-s − 6·47-s + 9·49-s − 6·51-s + 6·53-s − 4·57-s − 6·59-s + 6·61-s − 4·63-s − 4·69-s − 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s − 1.64·37-s + 0.160·39-s − 0.624·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.781·59-s + 0.768·61-s − 0.503·63-s − 0.481·69-s − 1.18·71-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
| \(L(1)\) |
\(\approx\) |
\(1.090373568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090373568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.26468204233815, −13.66373758327652, −13.13802603262385, −12.73954330299377, −12.26149328605295, −11.88648219073307, −11.29524253247485, −10.42071059846454, −10.28899196170075, −9.855144169349837, −9.113202757671358, −8.858159182978915, −7.814167377930731, −7.495084445358444, −6.880783817632744, −6.450195386511025, −5.757326302531472, −5.269365463859958, −4.929560744361054, −3.894567818745101, −3.223652915877501, −3.098682981813309, −2.070921376734914, −1.132392787668896, −0.4103653797572404,
0.4103653797572404, 1.132392787668896, 2.070921376734914, 3.098682981813309, 3.223652915877501, 3.894567818745101, 4.929560744361054, 5.269365463859958, 5.757326302531472, 6.450195386511025, 6.880783817632744, 7.495084445358444, 7.814167377930731, 8.858159182978915, 9.113202757671358, 9.855144169349837, 10.28899196170075, 10.42071059846454, 11.29524253247485, 11.88648219073307, 12.26149328605295, 12.73954330299377, 13.13802603262385, 13.66373758327652, 14.26468204233815
