L(s) = 1 | − 3-s + 4·7-s + 9-s − 5·11-s + 2·17-s − 6·19-s − 4·21-s − 3·23-s − 27-s − 10·29-s + 10·31-s + 5·33-s + 37-s − 6·41-s + 8·43-s − 4·47-s + 9·49-s − 2·51-s + 8·53-s + 6·57-s − 7·61-s + 4·63-s − 8·67-s + 3·69-s + 15·71-s + 9·73-s − 20·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.50·11-s + 0.485·17-s − 1.37·19-s − 0.872·21-s − 0.625·23-s − 0.192·27-s − 1.85·29-s + 1.79·31-s + 0.870·33-s + 0.164·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.280·51-s + 1.09·53-s + 0.794·57-s − 0.896·61-s + 0.503·63-s − 0.977·67-s + 0.361·69-s + 1.78·71-s + 1.05·73-s − 2.27·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−13.27023116341220, −12.75557778513206, −12.27504370113272, −11.92584181357017, −11.22667606373246, −11.01087827651706, −10.61956917718725, −10.08360349547010, −9.720840453896444, −8.906194660485485, −8.392517809276801, −7.987691040740455, −7.707598295977004, −7.161812797225099, −6.462188450072449, −5.916306269414993, −5.456663574875111, −5.032739614431773, −4.539625118318115, −4.109128500480556, −3.397807006527482, −2.499752730307287, −2.145495943882855, −1.553028407598638, −0.7423719515321683, 0,
0.7423719515321683, 1.553028407598638, 2.145495943882855, 2.499752730307287, 3.397807006527482, 4.109128500480556, 4.539625118318115, 5.032739614431773, 5.456663574875111, 5.916306269414993, 6.462188450072449, 7.161812797225099, 7.707598295977004, 7.987691040740455, 8.392517809276801, 8.906194660485485, 9.720840453896444, 10.08360349547010, 10.61956917718725, 11.01087827651706, 11.22667606373246, 11.92584181357017, 12.27504370113272, 12.75557778513206, 13.27023116341220