| L(s) = 1 | + 19.0·2-s + 138.·3-s − 148.·4-s − 952.·5-s + 2.63e3·6-s − 6.30e3·7-s − 1.25e4·8-s − 541.·9-s − 1.81e4·10-s − 1.81e4·11-s − 2.06e4·12-s − 2.24e4·13-s − 1.20e5·14-s − 1.31e5·15-s − 1.63e5·16-s − 1.03e4·18-s + 1.67e5·19-s + 1.41e5·20-s − 8.72e5·21-s − 3.45e5·22-s + 2.37e5·23-s − 1.74e6·24-s − 1.04e6·25-s − 4.28e5·26-s − 2.79e6·27-s + 9.39e5·28-s + 7.21e6·29-s + ⋯ |
| L(s) = 1 | + 0.842·2-s + 0.986·3-s − 0.290·4-s − 0.681·5-s + 0.830·6-s − 0.992·7-s − 1.08·8-s − 0.0275·9-s − 0.574·10-s − 0.373·11-s − 0.286·12-s − 0.218·13-s − 0.835·14-s − 0.672·15-s − 0.624·16-s − 0.0231·18-s + 0.295·19-s + 0.198·20-s − 0.978·21-s − 0.314·22-s + 0.176·23-s − 1.07·24-s − 0.535·25-s − 0.183·26-s − 1.01·27-s + 0.288·28-s + 1.89·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.773596313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773596313\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 19.0T + 512T^{2} \) |
| 3 | \( 1 - 138.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 952.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.30e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.81e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.24e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 1.67e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.37e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.21e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.65e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.52e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.44e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.86e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.86e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.78e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.48e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.27e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.79e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.94e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.73e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.06e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.52e8T + 7.60e17T^{2} \) |
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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
−10.01632730365529718407034837127, −9.172823002025505760785325906467, −8.408180144099526679203308386012, −7.40938832899090593848500619286, −6.19800116908631223341229912301, −5.10647315295638235722002100648, −3.90767063894357062326228469404, −3.29160285709364690702207460202, −2.45266947990418918782604183127, −0.46842421852742818334100384752,
0.46842421852742818334100384752, 2.45266947990418918782604183127, 3.29160285709364690702207460202, 3.90767063894357062326228469404, 5.10647315295638235722002100648, 6.19800116908631223341229912301, 7.40938832899090593848500619286, 8.408180144099526679203308386012, 9.172823002025505760785325906467, 10.01632730365529718407034837127
