| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 2·13-s + 14-s − 15-s + 16-s − 18-s + 4·19-s + 20-s + 21-s − 4·22-s + 24-s + 25-s + 2·26-s − 27-s − 28-s + 8·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.75549119193452, −13.61656918607879, −12.59898775237538, −12.35912661286142, −11.94533877115088, −11.48265596868213, −10.84356876465982, −10.44642189471619, −9.942582800562219, −9.464427687413139, −9.091267526352916, −8.638295083712930, −7.896408333073711, −7.283238620858089, −6.910002358843830, −6.427775634802427, −5.976008378624026, −5.230246252500822, −4.963509143163404, −3.972402150922681, −3.573930388376053, −2.790055223988227, −2.129472463118965, −1.398774743043141, −0.8907416082064658, 0,
0.8907416082064658, 1.398774743043141, 2.129472463118965, 2.790055223988227, 3.573930388376053, 3.972402150922681, 4.963509143163404, 5.230246252500822, 5.976008378624026, 6.427775634802427, 6.910002358843830, 7.283238620858089, 7.896408333073711, 8.638295083712930, 9.091267526352916, 9.464427687413139, 9.942582800562219, 10.44642189471619, 10.84356876465982, 11.48265596868213, 11.94533877115088, 12.35912661286142, 12.59898775237538, 13.61656918607879, 13.75549119193452
