L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s + 2·11-s − 12-s + 2·13-s + 2·14-s + 16-s − 4·17-s + 18-s + 2·19-s − 2·21-s + 2·22-s − 2·23-s − 24-s + 2·26-s − 27-s + 2·28-s + 6·29-s + 8·31-s + 32-s − 2·33-s − 4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.426·22-s − 0.417·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.348·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.96013615469785, −12.47691522383961, −12.08793897228245, −11.47520852626937, −11.30555058299188, −10.99251916814077, −10.32757848559718, −9.814967064243383, −9.458231557974653, −8.636931292937823, −8.333006533127035, −7.839805356626752, −7.265808666527008, −6.605581073438419, −6.383552027877314, −5.955280362707983, −5.282704867416633, −4.783008166530794, −4.413736785961766, −4.026876662160745, −3.304804091452881, −2.694706083984700, −2.127542382521809, −1.312904071861931, −1.072639447960371, 0,
1.072639447960371, 1.312904071861931, 2.127542382521809, 2.694706083984700, 3.304804091452881, 4.026876662160745, 4.413736785961766, 4.783008166530794, 5.282704867416633, 5.955280362707983, 6.383552027877314, 6.605581073438419, 7.265808666527008, 7.839805356626752, 8.333006533127035, 8.636931292937823, 9.458231557974653, 9.814967064243383, 10.32757848559718, 10.99251916814077, 11.30555058299188, 11.47520852626937, 12.08793897228245, 12.47691522383961, 12.96013615469785