| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 3·11-s − 12-s − 13-s − 15-s + 16-s − 3·17-s − 18-s − 2·19-s + 20-s + 3·22-s + 9·23-s + 24-s + 25-s + 26-s − 27-s − 6·29-s + 30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.639·22-s + 1.87·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.98984850766196, −15.62912399807367, −14.89613641625517, −14.70176985805436, −13.50780680883877, −13.26288246627034, −12.73689323915564, −12.07592698053096, −11.30728617390025, −11.02832870827873, −10.40907570492929, −9.933197303183110, −9.310940940911468, −8.705669955938250, −8.184736827513314, −7.312895301007312, −6.952385743985122, −6.323750395637407, −5.549429373401477, −5.082909371798277, −4.393403180621108, −3.344949656602297, −2.567642730058686, −1.919324556445208, −0.9375917694644956, 0,
0.9375917694644956, 1.919324556445208, 2.567642730058686, 3.344949656602297, 4.393403180621108, 5.082909371798277, 5.549429373401477, 6.323750395637407, 6.952385743985122, 7.312895301007312, 8.184736827513314, 8.705669955938250, 9.310940940911468, 9.933197303183110, 10.40907570492929, 11.02832870827873, 11.30728617390025, 12.07592698053096, 12.73689323915564, 13.26288246627034, 13.50780680883877, 14.70176985805436, 14.89613641625517, 15.62912399807367, 15.98984850766196
