| L(s) = 1 | + 0.997·2-s − 1.00·4-s + 3.25·5-s + 7-s − 2.99·8-s + 3.24·10-s + 1.13·11-s − 2.48·13-s + 0.997·14-s − 0.982·16-s + 2.73·17-s − 4.35·19-s − 3.26·20-s + 1.13·22-s − 1.12·23-s + 5.59·25-s − 2.47·26-s − 1.00·28-s + 4.75·29-s − 10.7·31-s + 5.01·32-s + 2.72·34-s + 3.25·35-s + 7.74·37-s − 4.34·38-s − 9.75·40-s − 3.17·41-s + ⋯ |
| L(s) = 1 | + 0.705·2-s − 0.502·4-s + 1.45·5-s + 0.377·7-s − 1.05·8-s + 1.02·10-s + 0.341·11-s − 0.688·13-s + 0.266·14-s − 0.245·16-s + 0.663·17-s − 0.999·19-s − 0.730·20-s + 0.240·22-s − 0.235·23-s + 1.11·25-s − 0.485·26-s − 0.189·28-s + 0.882·29-s − 1.93·31-s + 0.886·32-s + 0.467·34-s + 0.550·35-s + 1.27·37-s − 0.704·38-s − 1.54·40-s − 0.495·41-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
| \(L(1)\) |
\(\approx\) |
\(3.278577119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.278577119\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.997T + 2T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 7.74T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 3.00T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 - 9.91T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.87084436394767114210369821464, −6.89926378548708036928635589374, −6.24306712476359397095790589092, −5.54932749983803660066487259351, −5.22102860972886972297361651523, −4.34163526368576517967159057069, −3.68892901481598652964394527168, −2.58127803198257111311158192792, −2.02432769275052127796497640093, −0.811961768082086965715748030042,
0.811961768082086965715748030042, 2.02432769275052127796497640093, 2.58127803198257111311158192792, 3.68892901481598652964394527168, 4.34163526368576517967159057069, 5.22102860972886972297361651523, 5.54932749983803660066487259351, 6.24306712476359397095790589092, 6.89926378548708036928635589374, 7.87084436394767114210369821464
