| L(s) = 1 | − 2·3-s − 2·4-s − 14·7-s − 5·9-s + 4·12-s − 50·13-s + 4·16-s − 14·19-s + 28·21-s + 28·27-s + 28·28-s − 14·31-s + 10·36-s + 4·37-s + 100·39-s + 82·43-s − 8·48-s + 49·49-s + 100·52-s + 28·57-s − 2·61-s + 70·63-s − 8·64-s + 34·67-s − 140·73-s + 28·76-s − 116·79-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 1/2·4-s − 2·7-s − 5/9·9-s + 1/3·12-s − 3.84·13-s + 1/4·16-s − 0.736·19-s + 4/3·21-s + 1.03·27-s + 28-s − 0.451·31-s + 5/18·36-s + 4/37·37-s + 2.56·39-s + 1.90·43-s − 1/6·48-s + 49-s + 1.92·52-s + 0.491·57-s − 0.0327·61-s + 10/9·63-s − 1/8·64-s + 0.507·67-s − 1.91·73-s + 7/19·76-s − 1.46·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.007957933546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007957933546\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 170 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 70 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 410 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 118 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 41 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2090 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8282 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2590 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
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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
−12.80077935746623139225641811608, −12.45173475548742750593217424520, −12.30583293590574728480858659903, −11.71425854489656058054119170869, −10.98677574082474980001972357672, −10.26038770582260392884633175811, −9.995475528195160896285160164942, −9.465384406386950625864871437334, −9.256703092164790773438820338651, −8.426282351742969971479270964028, −7.51349113000620499237057728998, −7.19719697374105218027772967004, −6.58848367226181901897090713325, −5.94939396411308742126382586116, −5.32220201292438247460489817266, −4.76263833630216589244230829677, −4.03583413641834196898188828966, −2.82007134039990397847283642084, −2.59162756444145156550241095132, −0.05563988649986282565162414076,
0.05563988649986282565162414076, 2.59162756444145156550241095132, 2.82007134039990397847283642084, 4.03583413641834196898188828966, 4.76263833630216589244230829677, 5.32220201292438247460489817266, 5.94939396411308742126382586116, 6.58848367226181901897090713325, 7.19719697374105218027772967004, 7.51349113000620499237057728998, 8.426282351742969971479270964028, 9.256703092164790773438820338651, 9.465384406386950625864871437334, 9.995475528195160896285160164942, 10.26038770582260392884633175811, 10.98677574082474980001972357672, 11.71425854489656058054119170869, 12.30583293590574728480858659903, 12.45173475548742750593217424520, 12.80077935746623139225641811608
