L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s + 15-s + 16-s − 17-s − 18-s + 20-s − 22-s − 4·23-s − 24-s + 25-s + 27-s + 2·29-s − 30-s − 4·31-s − 32-s + 33-s + 34-s + 36-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 + 0.301·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.213·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.718·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 274890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 274890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.20167857167885, −12.38934790494223, −12.11868471221342, −11.62045315277545, −10.92932703453234, −10.71182025252495, −10.08544803910883, −9.692115431847323, −9.341348535832881, −8.827714147559031, −8.411434973021341, −7.988944028909833, −7.405851362481398, −7.057776525164097, −6.264130628693491, −6.237077663362465, −5.426985530359427, −4.832915197670197, −4.340209041045630, −3.541918148348189, −3.272468402504066, −2.515643250459249, −1.938499762678584, −1.616590946110937, −0.8077016361400987, 0,
0.8077016361400987, 1.616590946110937, 1.938499762678584, 2.515643250459249, 3.272468402504066, 3.541918148348189, 4.340209041045630, 4.832915197670197, 5.426985530359427, 6.237077663362465, 6.264130628693491, 7.057776525164097, 7.405851362481398, 7.988944028909833, 8.411434973021341, 8.827714147559031, 9.341348535832881, 9.692115431847323, 10.08544803910883, 10.71182025252495, 10.92932703453234, 11.62045315277545, 12.11868471221342, 12.38934790494223, 13.20167857167885