L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 11-s + 12-s + 6·13-s + 14-s + 16-s + 17-s − 18-s + 19-s − 21-s − 22-s − 2·23-s − 24-s − 6·26-s + 27-s − 28-s − 3·29-s + 7·31-s − 32-s + 33-s − 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.218·21-s − 0.213·22-s − 0.417·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + 1.25·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.395535041\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.395535041\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.91685379504494, −15.48808907284332, −14.71560384866513, −14.19986132314134, −13.64911291075755, −13.08595027835673, −12.55686454710430, −11.81521767258892, −11.26799338267568, −10.73819306620006, −10.08049764318908, −9.566581927375543, −8.950451135188051, −8.574649763175103, −7.861543660227702, −7.433409574900991, −6.570083884322697, −6.098716595500948, −5.535336151058859, −4.285162104311760, −3.876741136273569, −3.045424752152530, −2.416817999125426, −1.402339966231822, −0.7779306996804940,
0.7779306996804940, 1.402339966231822, 2.416817999125426, 3.045424752152530, 3.876741136273569, 4.285162104311760, 5.535336151058859, 6.098716595500948, 6.570083884322697, 7.433409574900991, 7.861543660227702, 8.574649763175103, 8.950451135188051, 9.566581927375543, 10.08049764318908, 10.73819306620006, 11.26799338267568, 11.81521767258892, 12.55686454710430, 13.08595027835673, 13.64911291075755, 14.19986132314134, 14.71560384866513, 15.48808907284332, 15.91685379504494