L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 11-s − 12-s − 13-s − 14-s + 16-s + 5·17-s + 18-s − 5·19-s + 21-s + 22-s + 23-s − 24-s − 5·25-s − 26-s − 27-s − 28-s + 9·29-s − 5·31-s + 32-s − 33-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 − 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 1.14·19-s + 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.67·29-s − 0.898·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.500847088\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.500847088\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.41855390735566, −16.05046023283788, −15.41139040375707, −14.78702332269516, −14.23798994704435, −13.75677436890663, −12.96883916651161, −12.35942444107372, −12.23677276780986, −11.41248104281667, −10.78724342960705, −10.27513648744374, −9.650479265740951, −8.938921642023874, −8.051392384464445, −7.459057298522546, −6.735621861962972, −6.152839276373167, −5.634693236927321, −4.889615581100854, −4.182440524760405, −3.547914241350204, −2.674843850277720, −1.774131681636237, −0.6826963345723457,
0.6826963345723457, 1.774131681636237, 2.674843850277720, 3.547914241350204, 4.182440524760405, 4.889615581100854, 5.634693236927321, 6.152839276373167, 6.735621861962972, 7.459057298522546, 8.051392384464445, 8.938921642023874, 9.650479265740951, 10.27513648744374, 10.78724342960705, 11.41248104281667, 12.23677276780986, 12.35942444107372, 12.96883916651161, 13.75677436890663, 14.23798994704435, 14.78702332269516, 15.41139040375707, 16.05046023283788, 16.41855390735566