| L(s) = 1 | − 9·9-s + 72·11-s − 280·19-s − 12·29-s − 32·31-s − 780·41-s − 338·49-s − 1.03e3·59-s − 116·61-s − 240·71-s + 2.33e3·79-s + 81·81-s + 3.18e3·89-s − 648·99-s + 1.59e3·101-s − 3.24e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.29e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.97·11-s − 3.38·19-s − 0.0768·29-s − 0.185·31-s − 2.97·41-s − 0.985·49-s − 2.27·59-s − 0.243·61-s − 0.401·71-s + 3.32·79-s + 1/9·81-s + 3.78·89-s − 0.657·99-s + 1.57·101-s − 2.85·109-s + 0.921·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.95·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.560174107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560174107\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 338 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4294 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3742 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12530 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 390 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156310 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 41182 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 284758 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 516 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 58 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 194138 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 360718 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1168 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 607750 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1590 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 p^{2} T^{2} + p^{6} T^{4} \) |
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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
−11.75307123070010315173154494862, −10.90725182594560341120921138472, −10.64920102579524409541995821930, −10.36467604784341250172957116832, −9.411476244455909597423906338065, −9.204288583641642825599473112711, −8.800520609734089177588183491482, −8.125320288630902359250080874159, −8.005441945174035097073673731921, −6.81047141401893679036521667879, −6.61932832013985628643186946535, −6.36392499259782491146598848347, −5.69545931301681462303955904762, −4.66716437551174447110892438396, −4.52385835277050934959864041572, −3.67082297408562190581801166016, −3.28836453152729999704062405703, −1.97244379491445672001452966247, −1.79576518050718373837789698804, −0.45303732462648160499282063648,
0.45303732462648160499282063648, 1.79576518050718373837789698804, 1.97244379491445672001452966247, 3.28836453152729999704062405703, 3.67082297408562190581801166016, 4.52385835277050934959864041572, 4.66716437551174447110892438396, 5.69545931301681462303955904762, 6.36392499259782491146598848347, 6.61932832013985628643186946535, 6.81047141401893679036521667879, 8.005441945174035097073673731921, 8.125320288630902359250080874159, 8.800520609734089177588183491482, 9.204288583641642825599473112711, 9.411476244455909597423906338065, 10.36467604784341250172957116832, 10.64920102579524409541995821930, 10.90725182594560341120921138472, 11.75307123070010315173154494862
