| L(s) = 1 | + 2.52·2-s + 4.40·4-s − 4.10·5-s + 7-s + 6.07·8-s − 10.3·10-s − 3.98·11-s + 4.99·13-s + 2.52·14-s + 6.56·16-s − 4.55·17-s − 5.49·19-s − 18.0·20-s − 10.0·22-s − 2.92·23-s + 11.8·25-s + 12.6·26-s + 4.40·28-s + 9.75·29-s + 7.97·31-s + 4.45·32-s − 11.5·34-s − 4.10·35-s + 4.83·37-s − 13.9·38-s − 24.9·40-s + 6.70·41-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.20·4-s − 1.83·5-s + 0.377·7-s + 2.14·8-s − 3.28·10-s − 1.20·11-s + 1.38·13-s + 0.676·14-s + 1.64·16-s − 1.10·17-s − 1.26·19-s − 4.04·20-s − 2.15·22-s − 0.609·23-s + 2.37·25-s + 2.47·26-s + 0.831·28-s + 1.81·29-s + 1.43·31-s + 0.788·32-s − 1.97·34-s − 0.694·35-s + 0.794·37-s − 2.25·38-s − 3.94·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.380317472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.380317472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 + 4.55T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 23 | \( 1 + 2.92T + 23T^{2} \) |
| 29 | \( 1 - 9.75T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 - 4.83T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 7.62T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 - 5.93T + 53T^{2} \) |
| 59 | \( 1 - 8.11T + 59T^{2} \) |
| 61 | \( 1 - 8.05T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 + 1.83T + 73T^{2} \) |
| 79 | \( 1 + 2.42T + 79T^{2} \) |
| 83 | \( 1 - 1.01T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.80708353429015988092799130225, −6.84065654219745125108471963598, −6.43553679060330026614542017858, −5.60847738780519900547838316572, −4.65193022557338960483475022544, −4.30288901859876262712816688518, −3.87550445934854627234929079370, −2.87608690697899123257187629003, −2.36123765584837233430385438081, −0.77000680457196929527628843689,
0.77000680457196929527628843689, 2.36123765584837233430385438081, 2.87608690697899123257187629003, 3.87550445934854627234929079370, 4.30288901859876262712816688518, 4.65193022557338960483475022544, 5.60847738780519900547838316572, 6.43553679060330026614542017858, 6.84065654219745125108471963598, 7.80708353429015988092799130225
