| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 5·11-s + 12-s + 16-s − 3·17-s − 18-s − 4·19-s + 5·22-s − 24-s + 27-s + 4·31-s − 32-s − 5·33-s + 3·34-s + 36-s + 4·38-s − 5·41-s − 7·43-s − 5·44-s + 4·47-s + 48-s − 7·49-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 − 1.50·11-s + 0.288·12-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s + 1.06·22-s − 0.204·24-s + 0.192·27-s + 0.718·31-s − 0.176·32-s − 0.870·33-s + 0.514·34-s + 1/6·36-s + 0.648·38-s − 0.780·41-s − 1.06·43-s − 0.753·44-s + 0.583·47-s + 0.144·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4563083835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4563083835\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.82811816470704, −13.58693336069906, −12.95679888760366, −12.65589428311236, −11.97621279999775, −11.42152356851768, −10.72557867872043, −10.56226500815277, −9.960558017837314, −9.491904637135862, −8.838922474076708, −8.429388861453149, −7.984380100573051, −7.587815254500403, −6.859233441802521, −6.462136005069365, −5.802265945724597, −5.048405083629727, −4.599669831107562, −3.885883272634423, −2.995998370910965, −2.727069026977734, −1.981257467343852, −1.422944277928503, −0.2263905953164112,
0.2263905953164112, 1.422944277928503, 1.981257467343852, 2.727069026977734, 2.995998370910965, 3.885883272634423, 4.599669831107562, 5.048405083629727, 5.802265945724597, 6.462136005069365, 6.859233441802521, 7.587815254500403, 7.984380100573051, 8.429388861453149, 8.838922474076708, 9.491904637135862, 9.960558017837314, 10.56226500815277, 10.72557867872043, 11.42152356851768, 11.97621279999775, 12.65589428311236, 12.95679888760366, 13.58693336069906, 13.82811816470704
