| L(s) = 1 | + 4·7-s − 9·9-s − 50·25-s + 92·31-s − 86·49-s − 36·63-s + 92·73-s + 284·79-s + 81·81-s + 4·97-s + 388·103-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + 173-s − 200·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4/7·7-s − 9-s − 2·25-s + 2.96·31-s − 1.75·49-s − 4/7·63-s + 1.26·73-s + 3.59·79-s + 81-s + 4/97·97-s + 3.76·103-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + 0.00578·173-s − 8/7·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.866253580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.866253580\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 146 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3214 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 1966 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 5906 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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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
−10.18001498609271687734124239396, −10.07894317476087984943378884364, −9.298124431561620862933162557262, −9.267560219003402184536503683054, −8.359960403707787313164502067115, −8.310756588004150683219898105583, −7.72813444060637483054792909800, −7.65643747070575351854773007716, −6.64362257394507595800497411810, −6.30416131674938740640170412013, −6.11170861738083795593455446879, −5.36963711045707406624635777561, −4.86336699844521943970010196486, −4.64603999232753337513652077992, −3.70784312821415518951851250659, −3.49077546506200036155167067615, −2.54094146412714703017665163071, −2.26818724145147922183916076725, −1.34919654385142630522390287015, −0.48709388338225333074117870277,
0.48709388338225333074117870277, 1.34919654385142630522390287015, 2.26818724145147922183916076725, 2.54094146412714703017665163071, 3.49077546506200036155167067615, 3.70784312821415518951851250659, 4.64603999232753337513652077992, 4.86336699844521943970010196486, 5.36963711045707406624635777561, 6.11170861738083795593455446879, 6.30416131674938740640170412013, 6.64362257394507595800497411810, 7.65643747070575351854773007716, 7.72813444060637483054792909800, 8.310756588004150683219898105583, 8.359960403707787313164502067115, 9.267560219003402184536503683054, 9.298124431561620862933162557262, 10.07894317476087984943378884364, 10.18001498609271687734124239396
