L(s) = 1 | − 2.56·2-s + 4.56·4-s − 3.56·5-s − 7-s − 6.56·8-s + 9.12·10-s − 3.12·11-s − 3.56·13-s + 2.56·14-s + 7.68·16-s − 4·17-s − 4·19-s − 16.2·20-s + 8·22-s + 0.438·23-s + 7.68·25-s + 9.12·26-s − 4.56·28-s − 2.43·29-s + 2.43·31-s − 6.56·32-s + 10.2·34-s + 3.56·35-s − 0.438·37-s + 10.2·38-s + 23.3·40-s − 7.12·41-s + ⋯ |
L(s) = 1 | − 1.81·2-s + 2.28·4-s − 1.59·5-s − 0.377·7-s − 2.31·8-s + 2.88·10-s − 0.941·11-s − 0.987·13-s + 0.684·14-s + 1.92·16-s − 0.970·17-s − 0.917·19-s − 3.63·20-s + 1.70·22-s + 0.0914·23-s + 1.53·25-s + 1.78·26-s − 0.862·28-s − 0.452·29-s + 0.437·31-s − 1.15·32-s + 1.75·34-s + 0.602·35-s − 0.0720·37-s + 1.66·38-s + 3.69·40-s − 1.11·41-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 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 0.438T + 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 + 0.438T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 - 8.68T + 53T^{2} \) |
| 59 | \( 1 - 8.68T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.73032211683421779928907743935, −7.01444652128041351223459933908, −6.72581441074997082813846840696, −5.56027267073244667372336750171, −4.55349549175864923199019321859, −3.77286360713724849663351949039, −2.69032327841900307545997980724, −2.19980399515283630398174074876, −0.66961197905301358082918100594, 0,
0.66961197905301358082918100594, 2.19980399515283630398174074876, 2.69032327841900307545997980724, 3.77286360713724849663351949039, 4.55349549175864923199019321859, 5.56027267073244667372336750171, 6.72581441074997082813846840696, 7.01444652128041351223459933908, 7.73032211683421779928907743935