| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6.37·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 12.7·10-s + 28.9·11-s + 12·12-s − 16.9·13-s − 14·14-s + 19.1·15-s + 16·16-s + 82.8·17-s + 18·18-s + 120.·19-s + 25.4·20-s − 21·21-s + 57.8·22-s − 23·23-s + 24·24-s − 84.3·25-s − 33.8·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.570·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.403·10-s + 0.792·11-s + 0.288·12-s − 0.361·13-s − 0.267·14-s + 0.329·15-s + 0.250·16-s + 1.18·17-s + 0.235·18-s + 1.45·19-s + 0.285·20-s − 0.218·21-s + 0.560·22-s − 0.208·23-s + 0.204·24-s − 0.675·25-s − 0.255·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.001614989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.001614989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 6.37T + 125T^{2} \) |
| 11 | \( 1 - 28.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 82.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 120.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 97.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 53.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 247.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 648.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 46.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 175.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 125.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 66.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 350.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 50.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−9.601103417338167043705087295640, −9.034798177803095224870055552595, −7.65013023563854435322036410483, −7.21506552185582344644048513804, −5.95209024318741741330424368027, −5.45105867824037600942215474759, −4.10002982041504772984073596254, −3.35691310538294869873462824640, −2.30855942129419605476313905320, −1.14286243439510759277341635411,
1.14286243439510759277341635411, 2.30855942129419605476313905320, 3.35691310538294869873462824640, 4.10002982041504772984073596254, 5.45105867824037600942215474759, 5.95209024318741741330424368027, 7.21506552185582344644048513804, 7.65013023563854435322036410483, 9.034798177803095224870055552595, 9.601103417338167043705087295640
