| L(s) = 1 | + 2-s − 3-s + 3·5-s − 6-s + 5·7-s − 8-s + 3·10-s + 11-s + 4·13-s + 5·14-s − 3·15-s − 16-s − 3·17-s − 2·19-s − 5·21-s + 22-s − 3·23-s + 24-s + 5·25-s + 4·26-s + 27-s − 12·29-s − 3·30-s + 4·31-s − 33-s − 3·34-s + 15·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1.34·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 0.948·10-s + 0.301·11-s + 1.10·13-s + 1.33·14-s − 0.774·15-s − 1/4·16-s − 0.727·17-s − 0.458·19-s − 1.09·21-s + 0.213·22-s − 0.625·23-s + 0.204·24-s + 25-s + 0.784·26-s + 0.192·27-s − 2.22·29-s − 0.547·30-s + 0.718·31-s − 0.174·33-s − 0.514·34-s + 2.53·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.127238854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.127238854\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.04597271497655279173690827563, −10.93654216649603626104166968522, −10.77998444165836686916566992690, −9.927907516137854598349173685749, −9.541855471454360321108817114676, −8.967276400106426869910190980417, −8.620128179545905334550509034037, −8.185130982465278066784869731568, −7.61737701820199576337490709246, −6.95901501898229785890438689766, −6.40988959433214787146001074595, −5.92248409012339033304400184717, −5.65081568096164035135449734555, −5.04441095709232249172197879272, −4.77496859845964703996739890820, −3.97342559205762001613961305927, −3.64220797129304656938683558994, −2.24133601191494122991144307934, −2.05750700543528053380001804863, −1.14550545010524135596556915406,
1.14550545010524135596556915406, 2.05750700543528053380001804863, 2.24133601191494122991144307934, 3.64220797129304656938683558994, 3.97342559205762001613961305927, 4.77496859845964703996739890820, 5.04441095709232249172197879272, 5.65081568096164035135449734555, 5.92248409012339033304400184717, 6.40988959433214787146001074595, 6.95901501898229785890438689766, 7.61737701820199576337490709246, 8.185130982465278066784869731568, 8.620128179545905334550509034037, 8.967276400106426869910190980417, 9.541855471454360321108817114676, 9.927907516137854598349173685749, 10.77998444165836686916566992690, 10.93654216649603626104166968522, 11.04597271497655279173690827563
