| L(s) = 1 | + 2·2-s + 3·3-s + 3·4-s + 6·6-s + 4·8-s + 6·9-s + 3·11-s + 9·12-s − 2·13-s + 5·16-s + 3·17-s + 12·18-s + 19-s + 6·22-s + 6·23-s + 12·24-s + 5·25-s − 4·26-s + 9·27-s − 6·29-s − 8·31-s + 6·32-s + 9·33-s + 6·34-s + 18·36-s + 4·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 3/2·4-s + 2.44·6-s + 1.41·8-s + 2·9-s + 0.904·11-s + 2.59·12-s − 0.554·13-s + 5/4·16-s + 0.727·17-s + 2.82·18-s + 0.229·19-s + 1.27·22-s + 1.25·23-s + 2.44·24-s + 25-s − 0.784·26-s + 1.73·27-s − 1.11·29-s − 1.43·31-s + 1.06·32-s + 1.56·33-s + 1.02·34-s + 3·36-s + 0.657·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.93790078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.93790078\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.13392241563349590928045444277, −9.973740158568599808165891965718, −9.413379541671919443440134293759, −9.187593106459345513386212339656, −8.522296638279674604940960935764, −8.310605276390878168261327514320, −7.72569731220448094664001112760, −7.16143522311662487577991187545, −6.98234093902369285393060980017, −6.74871591616088355860649585528, −5.71398731471673298124068200625, −5.61326356056976060480773403085, −4.70909731124347955715070624828, −4.62893596945869284610312352694, −3.78890154961404174102545011959, −3.52814471671964207722507487007, −3.02419902868192752170812323228, −2.66618941877582088921449385903, −1.74165258904569940954104414321, −1.42899325781132999807970473280,
1.42899325781132999807970473280, 1.74165258904569940954104414321, 2.66618941877582088921449385903, 3.02419902868192752170812323228, 3.52814471671964207722507487007, 3.78890154961404174102545011959, 4.62893596945869284610312352694, 4.70909731124347955715070624828, 5.61326356056976060480773403085, 5.71398731471673298124068200625, 6.74871591616088355860649585528, 6.98234093902369285393060980017, 7.16143522311662487577991187545, 7.72569731220448094664001112760, 8.310605276390878168261327514320, 8.522296638279674604940960935764, 9.187593106459345513386212339656, 9.413379541671919443440134293759, 9.973740158568599808165891965718, 10.13392241563349590928045444277
