L(s) = 1 | + 2-s + 3-s + 4-s − 3.91·5-s + 6-s − 4.71·7-s + 8-s + 9-s − 3.91·10-s − 4.45·11-s + 12-s − 3.78·13-s − 4.71·14-s − 3.91·15-s + 16-s + 17-s + 18-s − 5.42·19-s − 3.91·20-s − 4.71·21-s − 4.45·22-s − 5.61·23-s + 24-s + 10.3·25-s − 3.78·26-s + 27-s − 4.71·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.75·5-s + 0.408·6-s − 1.78·7-s + 0.353·8-s + 0.333·9-s − 1.23·10-s − 1.34·11-s + 0.288·12-s − 1.05·13-s − 1.26·14-s − 1.01·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.24·19-s − 0.876·20-s − 1.02·21-s − 0.950·22-s − 1.17·23-s + 0.204·24-s + 2.07·25-s − 0.743·26-s + 0.192·27-s − 0.891·28-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
\(L(1)\) |
\(\approx\) |
\(0.8115017107\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8115017107\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 19 | \( 1 + 5.42T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 - 8.21T + 29T^{2} \) |
| 31 | \( 1 - 0.794T + 31T^{2} \) |
| 37 | \( 1 + 1.18T + 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 - 1.69T + 47T^{2} \) |
| 53 | \( 1 + 7.69T + 53T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 6.44T + 71T^{2} \) |
| 73 | \( 1 - 6.22T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 3.99T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 + 5.12T + 97T^{2} \) |
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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
−7.979896559924062591988951847476, −7.36757438940157908680858599869, −6.75204175540081409350653637498, −6.04926262040827033943821262015, −4.94479308393322733788971619855, −4.26187667595159636109246348969, −3.65580272696965161331539934258, −2.88388570294999604344603517462, −2.46904552347930364810973830886, −0.37696900758322850829554453255,
0.37696900758322850829554453255, 2.46904552347930364810973830886, 2.88388570294999604344603517462, 3.65580272696965161331539934258, 4.26187667595159636109246348969, 4.94479308393322733788971619855, 6.04926262040827033943821262015, 6.75204175540081409350653637498, 7.36757438940157908680858599869, 7.979896559924062591988951847476