L(s) = 1 | + 28.6·2-s + 729·3-s − 7.37e3·4-s + 6.00e4·5-s + 2.08e4·6-s + 4.28e5·7-s − 4.45e5·8-s + 5.31e5·9-s + 1.71e6·10-s − 5.05e6·11-s − 5.37e6·12-s + 8.30e6·13-s + 1.22e7·14-s + 4.37e7·15-s + 4.76e7·16-s − 3.21e7·17-s + 1.52e7·18-s + 9.20e7·19-s − 4.42e8·20-s + 3.12e8·21-s − 1.44e8·22-s + 2.82e8·23-s − 3.24e8·24-s + 2.38e9·25-s + 2.37e8·26-s + 3.87e8·27-s − 3.15e9·28-s + ⋯ |
L(s) = 1 | + 0.316·2-s + 0.577·3-s − 0.900·4-s + 1.71·5-s + 0.182·6-s + 1.37·7-s − 0.600·8-s + 0.333·9-s + 0.543·10-s − 0.859·11-s − 0.519·12-s + 0.477·13-s + 0.434·14-s + 0.992·15-s + 0.710·16-s − 0.322·17-s + 0.105·18-s + 0.448·19-s − 1.54·20-s + 0.793·21-s − 0.271·22-s + 0.397·23-s − 0.346·24-s + 1.95·25-s + 0.150·26-s + 0.192·27-s − 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(4.962009555\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.962009555\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
good | 2 | \( 1 - 28.6T + 8.19e3T^{2} \) |
| 5 | \( 1 - 6.00e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 4.28e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.05e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 8.30e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 3.21e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 9.20e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 2.82e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.86e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 1.29e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 5.91e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.67e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 5.00e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 5.50e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 5.12e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 4.21e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 6.77e9T + 5.48e23T^{2} \) |
| 71 | \( 1 - 6.84e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.74e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 4.15e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.01e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 4.21e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 3.08e12T + 6.73e25T^{2} \) |
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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
−10.18374043432452673001887962938, −9.236954254963797357097021459286, −8.588733191138128259478108768530, −7.49367487351830048581576609423, −5.85281149375063562751705653060, −5.24577106347655488742999216615, −4.29281058649650045554737948497, −2.81705224412214429376779576017, −1.87828664533289205259380790556, −0.932097047379386479517097638137,
0.932097047379386479517097638137, 1.87828664533289205259380790556, 2.81705224412214429376779576017, 4.29281058649650045554737948497, 5.24577106347655488742999216615, 5.85281149375063562751705653060, 7.49367487351830048581576609423, 8.588733191138128259478108768530, 9.236954254963797357097021459286, 10.18374043432452673001887962938