L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 4·11-s + 12-s + 6·13-s + 2·14-s + 16-s + 6·17-s − 18-s + 2·19-s − 2·21-s − 4·22-s + 4·23-s − 24-s − 6·26-s + 27-s − 2·28-s − 8·29-s − 32-s + 4·33-s − 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.436·21-s − 0.852·22-s + 0.834·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.176·32-s + 0.696·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.160931905\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160931905\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.367418009148743603298870889502, −7.38424728412892850823436923230, −6.99075179408872469051123192550, −6.02549059183964819986862669375, −5.62820017784549046364141756867, −4.11645595686906429637606044719, −3.53401570603741256660153328185, −2.92928221029601801420686791171, −1.58715526849684412048385265037, −0.945378454453748230189584966731,
0.945378454453748230189584966731, 1.58715526849684412048385265037, 2.92928221029601801420686791171, 3.53401570603741256660153328185, 4.11645595686906429637606044719, 5.62820017784549046364141756867, 6.02549059183964819986862669375, 6.99075179408872469051123192550, 7.38424728412892850823436923230, 8.367418009148743603298870889502