| L(s) = 1 | − 2-s + 4-s − 2.82·5-s − 8-s − 3·9-s + 2.82·10-s + 4·11-s − 2.82·13-s + 16-s − 2.82·17-s + 3·18-s − 5.65·19-s − 2.82·20-s − 4·22-s + 3.00·25-s + 2.82·26-s − 29-s + 2.82·31-s − 32-s + 2.82·34-s − 3·36-s + 2·37-s + 5.65·38-s + 2.82·40-s + 2.82·41-s − 4·43-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.26·5-s − 0.353·8-s − 9-s + 0.894·10-s + 1.20·11-s − 0.784·13-s + 0.250·16-s − 0.685·17-s + 0.707·18-s − 1.29·19-s − 0.632·20-s − 0.852·22-s + 0.600·25-s + 0.554·26-s − 0.185·29-s + 0.508·31-s − 0.176·32-s + 0.485·34-s − 0.5·36-s + 0.328·37-s + 0.917·38-s + 0.447·40-s + 0.441·41-s − 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5520938682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5520938682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−8.814694932087859540482354971749, −8.103664248180547916827584197625, −7.44463153424167481111109347022, −6.63093717934388393725121519443, −6.00522186838979527912062788178, −4.70279036013702961671974542418, −4.01256311971922162003915813180, −3.05662959325665362084435039168, −2.02848517544519704744690436323, −0.48723641944338513758723498003,
0.48723641944338513758723498003, 2.02848517544519704744690436323, 3.05662959325665362084435039168, 4.01256311971922162003915813180, 4.70279036013702961671974542418, 6.00522186838979527912062788178, 6.63093717934388393725121519443, 7.44463153424167481111109347022, 8.103664248180547916827584197625, 8.814694932087859540482354971749
