| L(s) = 1 | − 1.90·2-s + 1.63·4-s − 1.97·5-s + 7-s + 0.698·8-s + 3.77·10-s + 5.17·11-s − 1.58·13-s − 1.90·14-s − 4.59·16-s + 8.13·17-s + 4.36·19-s − 3.23·20-s − 9.87·22-s + 7.89·23-s − 1.08·25-s + 3.02·26-s + 1.63·28-s − 2.09·29-s − 0.621·31-s + 7.36·32-s − 15.5·34-s − 1.97·35-s + 0.803·37-s − 8.31·38-s − 1.38·40-s + 2.54·41-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.816·4-s − 0.885·5-s + 0.377·7-s + 0.247·8-s + 1.19·10-s + 1.56·11-s − 0.440·13-s − 0.509·14-s − 1.14·16-s + 1.97·17-s + 1.00·19-s − 0.723·20-s − 2.10·22-s + 1.64·23-s − 0.216·25-s + 0.593·26-s + 0.308·28-s − 0.388·29-s − 0.111·31-s + 1.30·32-s − 2.65·34-s − 0.334·35-s + 0.132·37-s − 1.34·38-s − 0.218·40-s + 0.397·41-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
| \(L(1)\) |
\(\approx\) |
\(1.235405056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235405056\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.90T + 2T^{2} \) |
| 5 | \( 1 + 1.97T + 5T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 8.13T + 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 + 0.621T + 31T^{2} \) |
| 37 | \( 1 - 0.803T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 5.22T + 59T^{2} \) |
| 61 | \( 1 - 5.63T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 - 2.26T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.77570277334650306039558389578, −7.38704975149987825090071664936, −6.96711765902769652844073019730, −5.81259791079281358941204979125, −5.08181941011072507933544707084, −4.09818286259246100382186545021, −3.57813382211351059706159127834, −2.47018141019559096380149782930, −1.13653292908963729247847794266, −0.897676057611291859060084143837,
0.897676057611291859060084143837, 1.13653292908963729247847794266, 2.47018141019559096380149782930, 3.57813382211351059706159127834, 4.09818286259246100382186545021, 5.08181941011072507933544707084, 5.81259791079281358941204979125, 6.96711765902769652844073019730, 7.38704975149987825090071664936, 7.77570277334650306039558389578
