| L(s) = 1 | + 2.57·2-s − 3·3-s − 1.35·4-s − 18.9·5-s − 7.73·6-s − 7·7-s − 24.1·8-s + 9·9-s − 48.7·10-s − 13.2·11-s + 4.05·12-s + 1.04·13-s − 18.0·14-s + 56.7·15-s − 51.3·16-s − 23.3·17-s + 23.2·18-s + 18.4·19-s + 25.5·20-s + 21·21-s − 34.1·22-s − 23·23-s + 72.3·24-s + 232.·25-s + 2.69·26-s − 27·27-s + 9.45·28-s + ⋯ |
| L(s) = 1 | + 0.911·2-s − 0.577·3-s − 0.168·4-s − 1.69·5-s − 0.526·6-s − 0.377·7-s − 1.06·8-s + 0.333·9-s − 1.54·10-s − 0.363·11-s + 0.0974·12-s + 0.0223·13-s − 0.344·14-s + 0.976·15-s − 0.802·16-s − 0.332·17-s + 0.303·18-s + 0.222·19-s + 0.285·20-s + 0.218·21-s − 0.331·22-s − 0.208·23-s + 0.615·24-s + 1.85·25-s + 0.0203·26-s − 0.192·27-s + 0.0637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8006875089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8006875089\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 2.57T + 8T^{2} \) |
| 5 | \( 1 + 18.9T + 125T^{2} \) |
| 11 | \( 1 + 13.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.04T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 18.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 26.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 22.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 107.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 61.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 109.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 233.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 16.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 199.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 720.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 145.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 802.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 526.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.76e3T + 9.12e5T^{2} \) |
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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
−10.99893395562128104534229286930, −9.744673374468925762400006632436, −8.649979823090321690518189264577, −7.73311070869083859639487555023, −6.75755229746928584312465925247, −5.68186957430169233059926355964, −4.62435677784485490924444644659, −3.97431992422364974355069532861, −2.97736743962952245841752802573, −0.48508291905736385681593376658,
0.48508291905736385681593376658, 2.97736743962952245841752802573, 3.97431992422364974355069532861, 4.62435677784485490924444644659, 5.68186957430169233059926355964, 6.75755229746928584312465925247, 7.73311070869083859639487555023, 8.649979823090321690518189264577, 9.744673374468925762400006632436, 10.99893395562128104534229286930
