| L(s) = 1 | − 43.7·2-s + 83.5·3-s + 1.40e3·4-s − 712.·5-s − 3.65e3·6-s + 5.54e3·7-s − 3.89e4·8-s − 1.26e4·9-s + 3.11e4·10-s + 3.60e4·11-s + 1.17e5·12-s − 1.75e5·13-s − 2.42e5·14-s − 5.95e4·15-s + 9.84e5·16-s + 5.55e5·18-s − 5.12e5·19-s − 9.98e5·20-s + 4.63e5·21-s − 1.57e6·22-s − 1.41e6·23-s − 3.25e6·24-s − 1.44e6·25-s + 7.67e6·26-s − 2.70e6·27-s + 7.76e6·28-s − 6.23e6·29-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.595·3-s + 2.73·4-s − 0.509·5-s − 1.15·6-s + 0.872·7-s − 3.35·8-s − 0.644·9-s + 0.985·10-s + 0.743·11-s + 1.63·12-s − 1.70·13-s − 1.68·14-s − 0.303·15-s + 3.75·16-s + 1.24·18-s − 0.902·19-s − 1.39·20-s + 0.519·21-s − 1.43·22-s − 1.05·23-s − 2.00·24-s − 0.740·25-s + 3.29·26-s − 0.980·27-s + 2.38·28-s − 1.63·29-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
| \(L(5)\) |
\(\approx\) |
\(0.4099606389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4099606389\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 43.7T + 512T^{2} \) |
| 3 | \( 1 - 83.5T + 1.96e4T^{2} \) |
| 5 | \( 1 + 712.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.54e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.60e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.75e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.12e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.41e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.00e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.77e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.78e5T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.35e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.00e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.55e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.95e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.53e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.41e9T + 7.60e17T^{2} \) |
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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
−9.902913627961056010592479407476, −9.193322419161827540475130111076, −8.328270908134123265542750990819, −7.76140548563047452515989491854, −6.98324611532033645284866585113, −5.63648315845313463712632099230, −3.83693662208917860487103373737, −2.37209993878298944866706660632, −1.84967177180516146745348651168, −0.34964157972179207476644448935,
0.34964157972179207476644448935, 1.84967177180516146745348651168, 2.37209993878298944866706660632, 3.83693662208917860487103373737, 5.63648315845313463712632099230, 6.98324611532033645284866585113, 7.76140548563047452515989491854, 8.328270908134123265542750990819, 9.193322419161827540475130111076, 9.902913627961056010592479407476
