L(s) = 1 | + 1.11·3-s − 4.32·5-s − 1.76·9-s + 4.52·11-s − 0.118·13-s − 4.80·15-s + 1.64·17-s + 3.95·19-s − 3.93·23-s + 13.6·25-s − 5.29·27-s + 9.43·29-s − 6.16·31-s + 5.03·33-s − 2.15·37-s − 0.131·39-s − 9.51·41-s − 10.2·43-s + 7.61·45-s − 1.88·47-s + 1.83·51-s − 3.12·53-s − 19.5·55-s + 4.39·57-s − 4.58·59-s − 9.01·61-s + 0.511·65-s + ⋯ |
L(s) = 1 | + 0.642·3-s − 1.93·5-s − 0.587·9-s + 1.36·11-s − 0.0328·13-s − 1.24·15-s + 0.399·17-s + 0.907·19-s − 0.820·23-s + 2.73·25-s − 1.01·27-s + 1.75·29-s − 1.10·31-s + 0.876·33-s − 0.353·37-s − 0.0210·39-s − 1.48·41-s − 1.55·43-s + 1.13·45-s − 0.275·47-s + 0.256·51-s − 0.429·53-s − 2.63·55-s + 0.582·57-s − 0.597·59-s − 1.15·61-s + 0.0634·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.11T + 3T^{2} \) |
| 5 | \( 1 + 4.32T + 5T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 + 0.118T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + 3.93T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 + 6.16T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 + 9.51T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 + 9.01T + 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 0.862T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−8.404908107465441385675424911787, −7.83826318330307693690490898923, −7.10512836537857584480462790934, −6.36547136374590596263902579479, −5.12484546492654245453438597648, −4.25617857782538889935876847898, −3.46582937000251405969459259860, −3.09132073728271421539089495345, −1.43454940219164847804106619613, 0,
1.43454940219164847804106619613, 3.09132073728271421539089495345, 3.46582937000251405969459259860, 4.25617857782538889935876847898, 5.12484546492654245453438597648, 6.36547136374590596263902579479, 7.10512836537857584480462790934, 7.83826318330307693690490898923, 8.404908107465441385675424911787