L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s + 13-s + 16-s + 7·17-s + 18-s + 19-s + 4·22-s + 6·23-s + 24-s + 26-s + 27-s − 9·29-s − 3·31-s + 32-s + 4·33-s + 7·34-s + 36-s + 5·37-s + 38-s + 39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 0.229·19-s + 0.852·22-s + 1.25·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 1.67·29-s − 0.538·31-s + 0.176·32-s + 0.696·33-s + 1.20·34-s + 1/6·36-s + 0.821·37-s + 0.162·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.935332736\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.935332736\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.30966867626251, −13.05720701563109, −12.50041002919056, −12.12848970033011, −11.44532779587407, −11.22755954949921, −10.63969745714009, −9.923996165868778, −9.491050895781547, −9.199617615142405, −8.507960757004234, −7.960300943281034, −7.468693418179541, −7.025398797879909, −6.486604297485691, −5.906734785654315, −5.352846382300775, −4.939397815496259, −4.129587098794075, −3.640215335721275, −3.365796232548874, −2.727857885377633, −1.881243260812092, −1.386372891692362, −0.7483326694028744,
0.7483326694028744, 1.386372891692362, 1.881243260812092, 2.727857885377633, 3.365796232548874, 3.640215335721275, 4.129587098794075, 4.939397815496259, 5.352846382300775, 5.906734785654315, 6.486604297485691, 7.025398797879909, 7.468693418179541, 7.960300943281034, 8.507960757004234, 9.199617615142405, 9.491050895781547, 9.923996165868778, 10.63969745714009, 11.22755954949921, 11.44532779587407, 12.12848970033011, 12.50041002919056, 13.05720701563109, 13.30966867626251