| L(s) = 1 | − 2·3-s + 3·4-s + 3·9-s − 6·12-s + 5·16-s + 14·17-s + 12·23-s + 9·25-s − 4·27-s − 2·29-s + 9·36-s − 12·43-s − 10·48-s + 10·49-s − 28·51-s − 18·53-s + 2·61-s + 3·64-s + 42·68-s − 24·69-s − 18·75-s − 8·79-s + 5·81-s + 4·87-s + 36·92-s + 27·100-s − 6·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s + 9-s − 1.73·12-s + 5/4·16-s + 3.39·17-s + 2.50·23-s + 9/5·25-s − 0.769·27-s − 0.371·29-s + 3/2·36-s − 1.82·43-s − 1.44·48-s + 10/7·49-s − 3.92·51-s − 2.47·53-s + 0.256·61-s + 3/8·64-s + 5.09·68-s − 2.88·69-s − 2.07·75-s − 0.900·79-s + 5/9·81-s + 0.428·87-s + 3.75·92-s + 2.69·100-s − 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.211859914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.211859914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.98472859334651605966597579960, −10.88302241893883361078222358039, −10.24001098122455791523737200437, −10.23954221115180697276067613211, −9.368930806078042498823291254679, −9.162590430923933221910568357790, −8.080677528807436121356495576453, −8.010402834333492000945264505915, −7.16565704900569498187389351720, −7.09045611129576374140945211260, −6.61560568041594322329701925057, −6.08812833965906048320506583663, −5.37490673361024309327036264971, −5.32945675881602155587182282086, −4.74862985842187966766621402708, −3.67243749731344640759025038789, −3.08186222949977246792553401651, −2.79844164800277790581293115704, −1.31025962131053228163622230969, −1.24294828782970952351583060893,
1.24294828782970952351583060893, 1.31025962131053228163622230969, 2.79844164800277790581293115704, 3.08186222949977246792553401651, 3.67243749731344640759025038789, 4.74862985842187966766621402708, 5.32945675881602155587182282086, 5.37490673361024309327036264971, 6.08812833965906048320506583663, 6.61560568041594322329701925057, 7.09045611129576374140945211260, 7.16565704900569498187389351720, 8.010402834333492000945264505915, 8.080677528807436121356495576453, 9.162590430923933221910568357790, 9.368930806078042498823291254679, 10.23954221115180697276067613211, 10.24001098122455791523737200437, 10.88302241893883361078222358039, 10.98472859334651605966597579960
