L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 5·11-s + 12-s − 3·13-s + 14-s + 16-s + 2·17-s + 18-s + 3·19-s + 21-s + 5·22-s + 24-s − 3·26-s + 27-s + 28-s − 29-s + 32-s + 5·33-s + 2·34-s + 36-s + 4·37-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 + 1.50·11-s + 0.288·12-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.688·19-s + 0.218·21-s + 1.06·22-s + 0.204·24-s − 0.588·26-s + 0.192·27-s + 0.188·28-s − 0.185·29-s + 0.176·32-s + 0.870·33-s + 0.342·34-s + 1/6·36-s + 0.657·37-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
\(L(1)\) |
\(\approx\) |
\(6.970925137\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.970925137\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.09917978253417, −13.64651302976016, −13.08036463287258, −12.47806821535390, −11.98248836794757, −11.78994789298953, −11.11726876739205, −10.56932909619043, −9.931431936731776, −9.365323569428619, −9.144171929140763, −8.334339848083540, −7.814236397897965, −7.277874621686957, −6.916262670295415, −6.157911458744329, −5.738686109989728, −4.986401169296612, −4.452831523887649, −4.028018057057374, −3.285669169391260, −2.905429735366413, −2.024441184701248, −1.514513777856272, −0.7509409539551695,
0.7509409539551695, 1.514513777856272, 2.024441184701248, 2.905429735366413, 3.285669169391260, 4.028018057057374, 4.452831523887649, 4.986401169296612, 5.738686109989728, 6.157911458744329, 6.916262670295415, 7.277874621686957, 7.814236397897965, 8.334339848083540, 9.144171929140763, 9.365323569428619, 9.931431936731776, 10.56932909619043, 11.11726876739205, 11.78994789298953, 11.98248836794757, 12.47806821535390, 13.08036463287258, 13.64651302976016, 14.09917978253417