| L(s) = 1 | − 0.601·2-s − 1.63·4-s − 4.40·5-s − 7-s + 2.18·8-s + 2.64·10-s − 4.30·11-s − 1.91·13-s + 0.601·14-s + 1.96·16-s − 7.97·17-s + 2.00·19-s + 7.21·20-s + 2.58·22-s − 2.11·23-s + 14.4·25-s + 1.14·26-s + 1.63·28-s + 4.21·29-s − 2.87·31-s − 5.55·32-s + 4.79·34-s + 4.40·35-s − 2.54·37-s − 1.20·38-s − 9.63·40-s + 6.34·41-s + ⋯ |
| L(s) = 1 | − 0.425·2-s − 0.819·4-s − 1.96·5-s − 0.377·7-s + 0.773·8-s + 0.837·10-s − 1.29·11-s − 0.530·13-s + 0.160·14-s + 0.490·16-s − 1.93·17-s + 0.459·19-s + 1.61·20-s + 0.551·22-s − 0.440·23-s + 2.88·25-s + 0.225·26-s + 0.309·28-s + 0.782·29-s − 0.516·31-s − 0.981·32-s + 0.822·34-s + 0.744·35-s − 0.417·37-s − 0.195·38-s − 1.52·40-s + 0.991·41-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 | \( 1 + 0.601T + 2T^{2} \) |
| 5 | \( 1 + 4.40T + 5T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 + 1.91T + 13T^{2} \) |
| 17 | \( 1 + 7.97T + 17T^{2} \) |
| 19 | \( 1 - 2.00T + 19T^{2} \) |
| 23 | \( 1 + 2.11T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.18T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 8.44T + 61T^{2} \) |
| 67 | \( 1 + 7.52T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 3.45T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 5.58T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.46288927287599893838534530718, −7.29564898042771217933753155694, −6.20952910887249859195667007060, −5.03671723435585422662096110493, −4.62749399032898675373665265754, −4.00915186390741591542341408457, −3.22872222943560493838027635515, −2.33658292631517206702639369034, −0.65975577139268783243106306102, 0,
0.65975577139268783243106306102, 2.33658292631517206702639369034, 3.22872222943560493838027635515, 4.00915186390741591542341408457, 4.62749399032898675373665265754, 5.03671723435585422662096110493, 6.20952910887249859195667007060, 7.29564898042771217933753155694, 7.46288927287599893838534530718
