| L(s) = 1 | − 256·4-s + 382·5-s − 1.14e5·13-s + 6.55e4·16-s + 6.33e4·17-s − 9.77e4·20-s + 7.29e4·25-s − 1.27e6·29-s + 2.70e6·37-s + 7.95e6·41-s + 2.92e7·52-s − 3.90e7·61-s − 1.67e7·64-s − 4.36e7·65-s − 1.62e7·68-s − 6.99e7·73-s + 2.50e7·80-s − 4.30e7·81-s + 2.42e7·85-s − 2.43e8·89-s + 2.09e8·97-s − 1.86e7·100-s − 2.97e8·101-s − 9.79e6·109-s − 4.46e8·113-s + 3.25e8·116-s + 1.49e8·125-s + ⋯ |
| L(s) = 1 | − 4-s + 0.611·5-s − 3.99·13-s + 16-s + 0.758·17-s − 0.611·20-s + 0.186·25-s − 1.79·29-s + 1.44·37-s + 2.81·41-s + 3.99·52-s − 2.82·61-s − 64-s − 2.44·65-s − 0.758·68-s − 2.46·73-s + 0.611·80-s − 81-s + 0.463·85-s − 3.88·89-s + 2.36·97-s − 0.186·100-s − 2.86·101-s − 0.0694·109-s − 2.73·113-s + 1.79·116-s + 0.611·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2071201317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2071201317\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 + p^{8} T^{2} \) |
| 17 | $C_2$ | \( 1 - 63358 T + p^{8} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{16} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 1054 T + p^{8} T^{2} )( 1 + 672 T + p^{8} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 57120 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 137760 T + p^{8} T^{2} )( 1 + 1407838 T + p^{8} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3632160 T + p^{8} T^{2} )( 1 + 925922 T + p^{8} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4374720 T + p^{8} T^{2} )( 1 - 3577922 T + p^{8} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9620638 T + p^{8} T^{2} )( 1 + 9620638 T + p^{8} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 18369120 T + p^{8} T^{2} )( 1 + 20722082 T + p^{8} T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 15227520 T + p^{8} T^{2} )( 1 + 54717118 T + p^{8} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + p^{16} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 121779840 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 173379838 T + p^{8} T^{2} )( 1 - 35904960 T + p^{8} T^{2} ) \) |
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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
−13.25056779908350107138844332144, −12.77507683219964899959124516084, −12.38636579390873450399481670525, −11.98322467782528166197482926775, −11.09386486643921398527195062870, −10.28376969719466904175491422413, −9.774757474134491234584953170455, −9.440110519071073407855744780884, −9.175498876932068368829863120120, −7.87565994792195663638916537403, −7.58225677586843423436342446234, −7.10862103694657135755187426909, −5.70919916593096717810021448679, −5.57718824680961506706994646115, −4.56042469800932645559778965838, −4.36083747298825242082780075499, −2.88355382703562498348297563722, −2.47869293140621368068375370681, −1.37075507544726326988016261412, −0.15020856023829346369032359055,
0.15020856023829346369032359055, 1.37075507544726326988016261412, 2.47869293140621368068375370681, 2.88355382703562498348297563722, 4.36083747298825242082780075499, 4.56042469800932645559778965838, 5.57718824680961506706994646115, 5.70919916593096717810021448679, 7.10862103694657135755187426909, 7.58225677586843423436342446234, 7.87565994792195663638916537403, 9.175498876932068368829863120120, 9.440110519071073407855744780884, 9.774757474134491234584953170455, 10.28376969719466904175491422413, 11.09386486643921398527195062870, 11.98322467782528166197482926775, 12.38636579390873450399481670525, 12.77507683219964899959124516084, 13.25056779908350107138844332144
