L(s) = 1 | − 2-s − 3-s + 4-s − 2.25·5-s + 6-s + 0.744·7-s − 8-s + 9-s + 2.25·10-s + 2.33·11-s − 12-s + 1.46·13-s − 0.744·14-s + 2.25·15-s + 16-s − 17-s − 18-s + 6.67·19-s − 2.25·20-s − 0.744·21-s − 2.33·22-s − 3.70·23-s + 24-s + 0.0669·25-s − 1.46·26-s − 27-s + 0.744·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.00·5-s + 0.408·6-s + 0.281·7-s − 0.353·8-s + 0.333·9-s + 0.711·10-s + 0.703·11-s − 0.288·12-s + 0.407·13-s − 0.199·14-s + 0.581·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 1.53·19-s − 0.503·20-s − 0.162·21-s − 0.497·22-s − 0.772·23-s + 0.204·24-s + 0.0133·25-s − 0.288·26-s − 0.192·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9707110975\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9707110975\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 0.744T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 19 | \( 1 - 6.67T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 - 2.98T + 29T^{2} \) |
| 31 | \( 1 + 0.755T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 + 2.14T + 43T^{2} \) |
| 47 | \( 1 - 7.97T + 47T^{2} \) |
| 53 | \( 1 + 0.555T + 53T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 0.684T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 4.18T + 89T^{2} \) |
| 97 | \( 1 - 6.31T + 97T^{2} \) |
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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
−7.994320750167738947091165917941, −7.51198408399462883805563369449, −6.78939200435816115362291252057, −6.07332517669587487659600188322, −5.31513365300661917869810155001, −4.30573737684603631735455064865, −3.76519384519229137972431148793, −2.73434567353762631263410320455, −1.47211075368750667006714875011, −0.64279924297663149402705860424,
0.64279924297663149402705860424, 1.47211075368750667006714875011, 2.73434567353762631263410320455, 3.76519384519229137972431148793, 4.30573737684603631735455064865, 5.31513365300661917869810155001, 6.07332517669587487659600188322, 6.78939200435816115362291252057, 7.51198408399462883805563369449, 7.994320750167738947091165917941