| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 5·11-s − 12-s + 7·13-s − 14-s + 15-s + 16-s + 2·17-s − 18-s + 5·19-s − 20-s − 21-s − 5·22-s − 23-s + 24-s + 25-s − 7·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s + 1.94·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.14·19-s − 0.223·20-s − 0.218·21-s − 1.06·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 1.37·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.771147643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.771147643\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.57888597514325, −12.07575428930949, −11.57043596648032, −11.38804612318272, −11.04535532834290, −10.47263784085871, −10.03316515447834, −9.338713075925155, −9.008789654888414, −8.748818432089912, −8.042517926060155, −7.503581597064997, −7.327418257575873, −6.566487459719741, −6.145803909534269, −5.770545220131924, −5.316252875392507, −4.402131557003928, −4.023847822982323, −3.558648887155056, −3.090391810862151, −2.084391274270786, −1.507766506894572, −0.9617781028536850, −0.6823739010328901,
0.6823739010328901, 0.9617781028536850, 1.507766506894572, 2.084391274270786, 3.090391810862151, 3.558648887155056, 4.023847822982323, 4.402131557003928, 5.316252875392507, 5.770545220131924, 6.145803909534269, 6.566487459719741, 7.327418257575873, 7.503581597064997, 8.042517926060155, 8.748818432089912, 9.008789654888414, 9.338713075925155, 10.03316515447834, 10.47263784085871, 11.04535532834290, 11.38804612318272, 11.57043596648032, 12.07575428930949, 12.57888597514325
