| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 6·11-s − 12-s − 14-s − 15-s + 16-s + 6·17-s − 18-s + 20-s − 21-s − 6·22-s + 24-s + 25-s − 27-s + 28-s + 10·29-s + 30-s − 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s − 1.27·22-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s + 1.85·29-s + 0.182·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.206217432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.206217432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.64711742084205, −12.07359981061853, −11.70256699565715, −11.48498856119083, −10.86296888046169, −10.29311657802882, −9.994102047770032, −9.592594959917898, −9.056885084941573, −8.596968787562409, −8.197480322078978, −7.555165674680360, −7.046128954974392, −6.664043965794438, −6.178730867389758, −5.744380474779791, −5.223194043615995, −4.631195081779768, −4.025512803535449, −3.529461749223113, −2.886264709346894, −2.151888502091091, −1.541961084183078, −1.003523042122294, −0.6975309597230688,
0.6975309597230688, 1.003523042122294, 1.541961084183078, 2.151888502091091, 2.886264709346894, 3.529461749223113, 4.025512803535449, 4.631195081779768, 5.223194043615995, 5.744380474779791, 6.178730867389758, 6.664043965794438, 7.046128954974392, 7.555165674680360, 8.197480322078978, 8.596968787562409, 9.056885084941573, 9.592594959917898, 9.994102047770032, 10.29311657802882, 10.86296888046169, 11.48498856119083, 11.70256699565715, 12.07359981061853, 12.64711742084205
