| L(s) = 1 | − 3·2-s + 4-s − 12·5-s + 21·8-s + 36·10-s − 12·11-s + 61·13-s − 71·16-s − 117·17-s − 2·19-s − 12·20-s + 36·22-s + 75·23-s + 19·25-s − 183·26-s − 3·29-s − 263·31-s + 45·32-s + 351·34-s + 218·37-s + 6·38-s − 252·40-s − 246·41-s + 515·43-s − 12·44-s − 225·46-s + 318·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 1.07·5-s + 0.928·8-s + 1.13·10-s − 0.328·11-s + 1.30·13-s − 1.10·16-s − 1.66·17-s − 0.0241·19-s − 0.134·20-s + 0.348·22-s + 0.679·23-s + 0.151·25-s − 1.38·26-s − 0.0192·29-s − 1.52·31-s + 0.248·32-s + 1.77·34-s + 0.968·37-s + 0.0256·38-s − 0.996·40-s − 0.937·41-s + 1.82·43-s − 0.0411·44-s − 0.721·46-s + 0.986·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 61 T + p^{3} T^{2} \) |
| 17 | \( 1 + 117 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 - 75 T + p^{3} T^{2} \) |
| 29 | \( 1 + 3 T + p^{3} T^{2} \) |
| 31 | \( 1 + 263 T + p^{3} T^{2} \) |
| 37 | \( 1 - 218 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 515 T + p^{3} T^{2} \) |
| 47 | \( 1 - 318 T + p^{3} T^{2} \) |
| 53 | \( 1 - 459 T + p^{3} T^{2} \) |
| 59 | \( 1 + 255 T + p^{3} T^{2} \) |
| 61 | \( 1 - 862 T + p^{3} T^{2} \) |
| 67 | \( 1 - 479 T + p^{3} T^{2} \) |
| 71 | \( 1 - 117 T + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + 646 T + p^{3} T^{2} \) |
| 83 | \( 1 + 348 T + p^{3} T^{2} \) |
| 89 | \( 1 + 585 T + p^{3} T^{2} \) |
| 97 | \( 1 - 376 T + p^{3} T^{2} \) |
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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
−8.791316918030503655193325219680, −8.248610948712359586107558185212, −7.42608117180450678014250657743, −6.76460294582229747913343293884, −5.52169488366083904879389127301, −4.32692502261132237623568443614, −3.79031075207849273471713289306, −2.28963125162363263125680752646, −0.968309919254316825845353008722, 0,
0.968309919254316825845353008722, 2.28963125162363263125680752646, 3.79031075207849273471713289306, 4.32692502261132237623568443614, 5.52169488366083904879389127301, 6.76460294582229747913343293884, 7.42608117180450678014250657743, 8.248610948712359586107558185212, 8.791316918030503655193325219680
