| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 4·7-s + 8-s + 9-s + 10-s + 12-s − 4·14-s + 15-s + 16-s + 2·17-s + 18-s − 4·19-s + 20-s − 4·21-s + 24-s + 25-s + 27-s − 4·28-s + 2·29-s + 30-s + 32-s + 2·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.182·30-s + 0.176·32-s + 0.342·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 218010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.682137759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.682137759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.11316346921313, −12.64747927047642, −12.18653599034287, −11.88039528300463, −11.01211842650754, −10.58657314669190, −10.15653851421460, −9.770080489096117, −9.296911860941343, −8.654965474398106, −8.459800349086344, −7.585320500553863, −7.066582032968701, −6.814340049679858, −6.192773134157078, −5.744143060904927, −5.384931044479151, −4.484544452314690, −4.126738840986587, −3.552008151728869, −2.967749547532416, −2.698507427677271, −1.990625936537689, −1.348556845679224, −0.4336244415733150,
0.4336244415733150, 1.348556845679224, 1.990625936537689, 2.698507427677271, 2.967749547532416, 3.552008151728869, 4.126738840986587, 4.484544452314690, 5.384931044479151, 5.744143060904927, 6.192773134157078, 6.814340049679858, 7.066582032968701, 7.585320500553863, 8.459800349086344, 8.654965474398106, 9.296911860941343, 9.770080489096117, 10.15653851421460, 10.58657314669190, 11.01211842650754, 11.88039528300463, 12.18653599034287, 12.64747927047642, 13.11316346921313
