| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 3·13-s + 14-s + 15-s + 16-s − 3·17-s − 18-s + 6·19-s − 20-s + 21-s + 22-s − 8·23-s + 24-s + 25-s + 3·26-s − 27-s − 28-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 − 0.301·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s − 0.188·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.91684875181059, −12.13990770244934, −11.90195104562291, −11.76765297716230, −11.14048938835227, −10.56209683354947, −10.12491055188726, −9.706734716670331, −9.596790271845469, −8.590930041634996, −8.399201902772829, −7.887511169875976, −7.218434745563708, −7.032286019224163, −6.497754962961214, −5.916528929706567, −5.388759256966479, −4.967413550279329, −4.264377221683841, −3.873278650257096, −2.959914466623393, −2.801315859810931, −1.854676406396673, −1.426567926412195, −0.4741764209994895, 0,
0.4741764209994895, 1.426567926412195, 1.854676406396673, 2.801315859810931, 2.959914466623393, 3.873278650257096, 4.264377221683841, 4.967413550279329, 5.388759256966479, 5.916528929706567, 6.497754962961214, 7.032286019224163, 7.218434745563708, 7.887511169875976, 8.399201902772829, 8.590930041634996, 9.596790271845469, 9.706734716670331, 10.12491055188726, 10.56209683354947, 11.14048938835227, 11.76765297716230, 11.90195104562291, 12.13990770244934, 12.91684875181059
