| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s + 2·13-s + 14-s − 16-s + 2·17-s − 6·19-s + 8·23-s − 5·25-s + 2·26-s − 28-s − 6·29-s − 2·31-s + 5·32-s + 2·34-s − 6·37-s − 6·38-s + 6·41-s + 4·43-s + 8·46-s − 2·47-s + 49-s − 5·50-s − 2·52-s − 2·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 1.37·19-s + 1.66·23-s − 25-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.359·31-s + 0.883·32-s + 0.342·34-s − 0.986·37-s − 0.973·38-s + 0.937·41-s + 0.609·43-s + 1.17·46-s − 0.291·47-s + 1/7·49-s − 0.707·50-s − 0.277·52-s − 0.274·53-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 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.44661457647161313893175524400, −6.68860864153631463965460824339, −5.83045828728348006060529191533, −5.44007169024062456318381662518, −4.59235641384718959277876291646, −3.98531292472348012307438964815, −3.34288681246265157218594770715, −2.38430741114076217972935825614, −1.29783714916541707932782890696, 0,
1.29783714916541707932782890696, 2.38430741114076217972935825614, 3.34288681246265157218594770715, 3.98531292472348012307438964815, 4.59235641384718959277876291646, 5.44007169024062456318381662518, 5.83045828728348006060529191533, 6.68860864153631463965460824339, 7.44661457647161313893175524400
