L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 0.267·7-s + 8-s + 9-s + 10-s + 12-s + 3·13-s + 0.267·14-s + 15-s + 16-s + 3.73·17-s + 18-s + 2.46·19-s + 20-s + 0.267·21-s − 1.26·23-s + 24-s + 25-s + 3·26-s + 27-s + 0.267·28-s − 3·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.101·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.288·12-s + 0.832·13-s + 0.0716·14-s + 0.258·15-s + 0.250·16-s + 0.905·17-s + 0.235·18-s + 0.565·19-s + 0.223·20-s + 0.0584·21-s − 0.264·23-s + 0.204·24-s + 0.200·25-s + 0.588·26-s + 0.192·27-s + 0.0506·28-s − 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.548981305\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.548981305\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 0.267T + 7T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 + 8.73T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 4.53T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 + 4.19T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 9.39T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.394268099470493082402590767464, −7.81703851237210618467699636495, −7.02546373954685779334712628009, −6.17863995950606854690903728613, −5.54518407433964995818255149902, −4.73722295130883785821621129594, −3.73239373445252563354893589814, −3.18675843951102880262587063785, −2.13428281247480610256209125136, −1.21369092461611589623152014380,
1.21369092461611589623152014380, 2.13428281247480610256209125136, 3.18675843951102880262587063785, 3.73239373445252563354893589814, 4.73722295130883785821621129594, 5.54518407433964995818255149902, 6.17863995950606854690903728613, 7.02546373954685779334712628009, 7.81703851237210618467699636495, 8.394268099470493082402590767464