L(s) = 1 | − 2-s − 7·3-s − 7·4-s + 7·6-s − 6·7-s + 15·8-s + 22·9-s − 43·11-s + 49·12-s + 28·13-s + 6·14-s + 41·16-s − 91·17-s − 22·18-s − 35·19-s + 42·21-s + 43·22-s − 162·23-s − 105·24-s − 28·26-s + 35·27-s + 42·28-s + 160·29-s + 42·31-s − 161·32-s + 301·33-s + 91·34-s + ⋯ |
L(s) = 1 | − 0.353·2-s − 1.34·3-s − 7/8·4-s + 0.476·6-s − 0.323·7-s + 0.662·8-s + 0.814·9-s − 1.17·11-s + 1.17·12-s + 0.597·13-s + 0.114·14-s + 0.640·16-s − 1.29·17-s − 0.288·18-s − 0.422·19-s + 0.436·21-s + 0.416·22-s − 1.46·23-s − 0.893·24-s − 0.211·26-s + 0.249·27-s + 0.283·28-s + 1.02·29-s + 0.243·31-s − 0.889·32-s + 1.58·33-s + 0.459·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 43 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 91 T + p^{3} T^{2} \) |
| 19 | \( 1 + 35 T + p^{3} T^{2} \) |
| 23 | \( 1 + 162 T + p^{3} T^{2} \) |
| 29 | \( 1 - 160 T + p^{3} T^{2} \) |
| 31 | \( 1 - 42 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 203 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 196 T + p^{3} T^{2} \) |
| 53 | \( 1 + 82 T + p^{3} T^{2} \) |
| 59 | \( 1 + 280 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 141 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 - 763 T + p^{3} T^{2} \) |
| 79 | \( 1 - 510 T + p^{3} T^{2} \) |
| 83 | \( 1 + 777 T + p^{3} T^{2} \) |
| 89 | \( 1 + 945 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1246 T + p^{3} T^{2} \) |
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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
−16.69961773350237799906472622064, −15.67093204476688056300260536922, −13.66966136312398800619567249168, −12.61757920584656200574446566405, −11.07012420934829748421911361250, −10.01294566678951525338184563642, −8.276840511944075708942991995655, −6.21043870300283489101869711821, −4.69045001633665639689119387631, 0,
4.69045001633665639689119387631, 6.21043870300283489101869711821, 8.276840511944075708942991995655, 10.01294566678951525338184563642, 11.07012420934829748421911361250, 12.61757920584656200574446566405, 13.66966136312398800619567249168, 15.67093204476688056300260536922, 16.69961773350237799906472622064