| L(s) = 1 | + 2-s − 3-s + 4-s + 3.05·5-s − 6-s + 8-s + 9-s + 3.05·10-s + 0.0875·11-s − 12-s + 2.29·13-s − 3.05·15-s + 16-s + 6.83·17-s + 18-s − 19-s + 3.05·20-s + 0.0875·22-s − 0.451·23-s − 24-s + 4.32·25-s + 2.29·26-s − 27-s + 9.78·29-s − 3.05·30-s − 6.48·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.36·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.965·10-s + 0.0264·11-s − 0.288·12-s + 0.635·13-s − 0.788·15-s + 0.250·16-s + 1.65·17-s + 0.235·18-s − 0.229·19-s + 0.682·20-s + 0.0186·22-s − 0.0941·23-s − 0.204·24-s + 0.865·25-s + 0.449·26-s − 0.192·27-s + 1.81·29-s − 0.557·30-s − 1.16·31-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
| \(L(1)\) |
\(\approx\) |
\(3.930678274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.930678274\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.05T + 5T^{2} \) |
| 11 | \( 1 - 0.0875T + 11T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 - 6.83T + 17T^{2} \) |
| 23 | \( 1 + 0.451T + 23T^{2} \) |
| 29 | \( 1 - 9.78T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 2.85T + 37T^{2} \) |
| 41 | \( 1 + 0.960T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 3.07T + 59T^{2} \) |
| 61 | \( 1 - 4.54T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 1.76T + 73T^{2} \) |
| 79 | \( 1 + 1.62T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 + 0.176T + 97T^{2} \) |
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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
−8.063785625796161816729257045657, −7.14995047941441001336433467349, −6.46491397783802737452480934106, −5.83023170545131478538982353927, −5.45213181603235986824689297996, −4.68947147682269440059547083996, −3.69705062622968668316729108699, −2.87293347465536735259532927647, −1.84576472733671239671714685536, −1.06582849682692811140751420909,
1.06582849682692811140751420909, 1.84576472733671239671714685536, 2.87293347465536735259532927647, 3.69705062622968668316729108699, 4.68947147682269440059547083996, 5.45213181603235986824689297996, 5.83023170545131478538982353927, 6.46491397783802737452480934106, 7.14995047941441001336433467349, 8.063785625796161816729257045657
