| L(s) = 1 | − 3-s − 6·5-s + 6·11-s + 6·15-s + 5·19-s − 12·23-s + 19·25-s + 27-s − 5·31-s − 6·33-s + 11·37-s − 6·47-s − 12·53-s − 36·55-s − 5·57-s + 12·59-s + 24·61-s − 15·67-s + 12·69-s + 9·73-s − 19·75-s − 21·79-s − 81-s − 36·83-s − 12·89-s + 5·93-s − 30·95-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2.68·5-s + 1.80·11-s + 1.54·15-s + 1.14·19-s − 2.50·23-s + 19/5·25-s + 0.192·27-s − 0.898·31-s − 1.04·33-s + 1.80·37-s − 0.875·47-s − 1.64·53-s − 4.85·55-s − 0.662·57-s + 1.56·59-s + 3.07·61-s − 1.83·67-s + 1.44·69-s + 1.05·73-s − 2.19·75-s − 2.36·79-s − 1/9·81-s − 3.95·83-s − 1.27·89-s + 0.518·93-s − 3.07·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2997262932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2997262932\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.671698580830243451694689247146, −8.528595555660986011144992333201, −8.449942892896759581205198928803, −7.88629172481279021040580104344, −7.78290243696478170870578388742, −7.17041631603365486434514503538, −7.06711350784399495827442739540, −6.48775595233884848843344908526, −6.20002311406581522914434435362, −5.47641920194200923920322113295, −5.42210625503084125942408110976, −4.36821327415526520552638126455, −4.35880064904406542696236399975, −3.84715448270943117777029326617, −3.84558346434773473541643891978, −3.12720229769520223265923479699, −2.63638186442779382525477860321, −1.59631233487813131159050231603, −1.12382345083544906122864893634, −0.22836289856137411672963545183,
0.22836289856137411672963545183, 1.12382345083544906122864893634, 1.59631233487813131159050231603, 2.63638186442779382525477860321, 3.12720229769520223265923479699, 3.84558346434773473541643891978, 3.84715448270943117777029326617, 4.35880064904406542696236399975, 4.36821327415526520552638126455, 5.42210625503084125942408110976, 5.47641920194200923920322113295, 6.20002311406581522914434435362, 6.48775595233884848843344908526, 7.06711350784399495827442739540, 7.17041631603365486434514503538, 7.78290243696478170870578388742, 7.88629172481279021040580104344, 8.449942892896759581205198928803, 8.528595555660986011144992333201, 9.671698580830243451694689247146
