L(s) = 1 | − 0.365·2-s − 1.86·4-s + 3.76·5-s − 2.92·7-s + 1.41·8-s − 1.37·10-s − 3.39·11-s + 1.65·13-s + 1.06·14-s + 3.21·16-s − 1.97·17-s − 5.94·19-s − 7.02·20-s + 1.24·22-s − 5.19·23-s + 9.15·25-s − 0.606·26-s + 5.45·28-s + 3.06·29-s + 6.78·31-s − 4.00·32-s + 0.722·34-s − 10.9·35-s + 1.04·37-s + 2.17·38-s + 5.32·40-s − 4.30·41-s + ⋯ |
L(s) = 1 | − 0.258·2-s − 0.933·4-s + 1.68·5-s − 1.10·7-s + 0.499·8-s − 0.435·10-s − 1.02·11-s + 0.460·13-s + 0.285·14-s + 0.803·16-s − 0.478·17-s − 1.36·19-s − 1.56·20-s + 0.264·22-s − 1.08·23-s + 1.83·25-s − 0.119·26-s + 1.03·28-s + 0.569·29-s + 1.21·31-s − 0.707·32-s + 0.123·34-s − 1.85·35-s + 0.171·37-s + 0.352·38-s + 0.841·40-s − 0.672·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.258512824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.258512824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 2 | \( 1 + 0.365T + 2T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 + 3.39T + 11T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 + 1.97T + 17T^{2} \) |
| 19 | \( 1 + 5.94T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 - 6.88T + 43T^{2} \) |
| 47 | \( 1 - 7.18T + 47T^{2} \) |
| 53 | \( 1 - 6.75T + 53T^{2} \) |
| 59 | \( 1 - 8.13T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 6.26T + 67T^{2} \) |
| 71 | \( 1 + 3.91T + 71T^{2} \) |
| 73 | \( 1 + 9.32T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.708150099093178084350538037071, −7.920725941575478643504482927783, −6.76235416045722014311576342752, −6.13629590168666747921224103409, −5.63971642783335600035700699444, −4.74795476965288286021323629724, −3.91126184007833321566558443403, −2.73999584248335936050996202316, −2.04967301856245707490895317510, −0.65542041145083084047414349573,
0.65542041145083084047414349573, 2.04967301856245707490895317510, 2.73999584248335936050996202316, 3.91126184007833321566558443403, 4.74795476965288286021323629724, 5.63971642783335600035700699444, 6.13629590168666747921224103409, 6.76235416045722014311576342752, 7.920725941575478643504482927783, 8.708150099093178084350538037071