| L(s) = 1 | + 2-s − 3-s + 4-s − 3.15·5-s − 6-s + 8-s + 9-s − 3.15·10-s − 6.08·11-s − 12-s − 1.70·13-s + 3.15·15-s + 16-s − 6.58·17-s + 18-s − 19-s − 3.15·20-s − 6.08·22-s − 5.06·23-s − 24-s + 4.94·25-s − 1.70·26-s − 27-s + 7.52·29-s + 3.15·30-s − 7.99·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.40·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.997·10-s − 1.83·11-s − 0.288·12-s − 0.473·13-s + 0.814·15-s + 0.250·16-s − 1.59·17-s + 0.235·18-s − 0.229·19-s − 0.704·20-s − 1.29·22-s − 1.05·23-s − 0.204·24-s + 0.988·25-s − 0.334·26-s − 0.192·27-s + 1.39·29-s + 0.575·30-s − 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6878383268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6878383268\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.15T + 5T^{2} \) |
| 11 | \( 1 + 6.08T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 + 6.58T + 17T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 - 7.52T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 4.03T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 6.81T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 + 2.42T + 67T^{2} \) |
| 71 | \( 1 + 0.0450T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 1.62T + 79T^{2} \) |
| 83 | \( 1 + 7.81T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.068265607985451065347139517119, −7.29745468240602565545981123747, −6.80972886974137574425546291716, −5.84603235218977580885288046459, −5.13315662979353188286458752121, −4.44578365352615273181546359782, −3.97814471581444752431886667555, −2.86313301777521331499930748331, −2.16139919142620796768019817315, −0.38012907181428562193576513315,
0.38012907181428562193576513315, 2.16139919142620796768019817315, 2.86313301777521331499930748331, 3.97814471581444752431886667555, 4.44578365352615273181546359782, 5.13315662979353188286458752121, 5.84603235218977580885288046459, 6.80972886974137574425546291716, 7.29745468240602565545981123747, 8.068265607985451065347139517119
