L(s) = 1 | − 0.308·2-s − 1.90·4-s + 1.85·5-s + 7-s + 1.20·8-s − 0.572·10-s − 1.15·11-s − 6.94·13-s − 0.308·14-s + 3.43·16-s − 5.72·17-s + 0.736·19-s − 3.53·20-s + 0.357·22-s − 8.44·23-s − 1.55·25-s + 2.14·26-s − 1.90·28-s − 2.24·29-s − 2.94·31-s − 3.46·32-s + 1.76·34-s + 1.85·35-s − 3.40·37-s − 0.227·38-s + 2.23·40-s + 8.37·41-s + ⋯ |
L(s) = 1 | − 0.218·2-s − 0.952·4-s + 0.829·5-s + 0.377·7-s + 0.425·8-s − 0.180·10-s − 0.349·11-s − 1.92·13-s − 0.0824·14-s + 0.859·16-s − 1.38·17-s + 0.168·19-s − 0.790·20-s + 0.0761·22-s − 1.76·23-s − 0.311·25-s + 0.420·26-s − 0.359·28-s − 0.415·29-s − 0.529·31-s − 0.613·32-s + 0.303·34-s + 0.313·35-s − 0.559·37-s − 0.0368·38-s + 0.353·40-s + 1.30·41-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
\(L(1)\) |
\(\approx\) |
\(0.8530998865\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8530998865\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.308T + 2T^{2} \) |
| 5 | \( 1 - 1.85T + 5T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 + 6.94T + 13T^{2} \) |
| 17 | \( 1 + 5.72T + 17T^{2} \) |
| 19 | \( 1 - 0.736T + 19T^{2} \) |
| 23 | \( 1 + 8.44T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 + 3.40T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 - 3.88T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 - 9.94T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 - 4.14T + 73T^{2} \) |
| 79 | \( 1 - 6.66T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 0.0324T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.83782532554539910248707509340, −7.32824731923718396257690171845, −6.42583267185341719281461507070, −5.54865913881906131946362079191, −5.08707371310201180707581572561, −4.41391260565268210145719949904, −3.66906408854760522044712040346, −2.23477908433073843116036238355, −2.06108345431833511102714894545, −0.45154462553574730063744338948,
0.45154462553574730063744338948, 2.06108345431833511102714894545, 2.23477908433073843116036238355, 3.66906408854760522044712040346, 4.41391260565268210145719949904, 5.08707371310201180707581572561, 5.54865913881906131946362079191, 6.42583267185341719281461507070, 7.32824731923718396257690171845, 7.83782532554539910248707509340