| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s − 2·9-s + 3·10-s + 12-s − 4·13-s + 14-s − 3·15-s + 16-s + 17-s + 2·18-s − 19-s − 3·20-s − 21-s + 6·23-s − 24-s + 4·25-s + 4·26-s − 5·27-s − 28-s − 9·29-s + 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.288·12-s − 1.10·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.229·19-s − 0.670·20-s − 0.218·21-s + 1.25·23-s − 0.204·24-s + 4/5·25-s + 0.784·26-s − 0.962·27-s − 0.188·28-s − 1.67·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7419796791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7419796791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.147378894428641144239212421871, −8.303820063650154077028632300707, −7.69939227626095242902463170062, −7.22288997848434944347871045634, −6.19148447974003595546337038208, −5.08057784576082493771015402336, −4.00858616572041932413278904611, −3.13276903562542106982684702493, −2.37933903207009454927787074561, −0.58914584796902051877888260181,
0.58914584796902051877888260181, 2.37933903207009454927787074561, 3.13276903562542106982684702493, 4.00858616572041932413278904611, 5.08057784576082493771015402336, 6.19148447974003595546337038208, 7.22288997848434944347871045634, 7.69939227626095242902463170062, 8.303820063650154077028632300707, 9.147378894428641144239212421871
