| L(s) = 1 | + 0.455·2-s − 1.79·4-s + 0.749·5-s − 7-s − 1.72·8-s + 0.341·10-s + 1.95·11-s + 2.51·13-s − 0.455·14-s + 2.79·16-s + 1.75·17-s − 3.76·19-s − 1.34·20-s + 0.892·22-s − 7.74·23-s − 4.43·25-s + 1.14·26-s + 1.79·28-s − 1.80·29-s + 7.78·31-s + 4.72·32-s + 0.800·34-s − 0.749·35-s + 2.63·37-s − 1.71·38-s − 1.29·40-s + 8.09·41-s + ⋯ |
| L(s) = 1 | + 0.322·2-s − 0.896·4-s + 0.335·5-s − 0.377·7-s − 0.610·8-s + 0.107·10-s + 0.590·11-s + 0.698·13-s − 0.121·14-s + 0.699·16-s + 0.426·17-s − 0.864·19-s − 0.300·20-s + 0.190·22-s − 1.61·23-s − 0.887·25-s + 0.224·26-s + 0.338·28-s − 0.335·29-s + 1.39·31-s + 0.835·32-s + 0.137·34-s − 0.126·35-s + 0.432·37-s − 0.278·38-s − 0.204·40-s + 1.26·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.455T + 2T^{2} \) |
| 5 | \( 1 - 0.749T + 5T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 7.74T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 - 8.09T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.25T + 47T^{2} \) |
| 53 | \( 1 + 0.297T + 53T^{2} \) |
| 59 | \( 1 + 0.440T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 1.44T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 + 9.80T + 97T^{2} \) |
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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
−7.64384975404116236122878503036, −6.40695490565605946479056179250, −6.15303725132182317964935647530, −5.49019202760972063022855609151, −4.47907407792383497025107816151, −4.00253284296947631846217776622, −3.33454229544694588861006883496, −2.27466698743818980821926612727, −1.20973251092890298222943111274, 0,
1.20973251092890298222943111274, 2.27466698743818980821926612727, 3.33454229544694588861006883496, 4.00253284296947631846217776622, 4.47907407792383497025107816151, 5.49019202760972063022855609151, 6.15303725132182317964935647530, 6.40695490565605946479056179250, 7.64384975404116236122878503036
