| L(s) = 1 | + 15·4-s + 86·11-s + 161·16-s + 70·19-s + 320·29-s + 84·31-s + 406·41-s + 1.29e3·44-s + 650·49-s − 560·59-s − 1.03e3·61-s + 1.45e3·64-s − 824·71-s + 1.05e3·76-s − 1.02e3·79-s − 1.89e3·89-s − 2.60e3·101-s − 2.14e3·109-s + 4.80e3·116-s + 2.88e3·121-s + 1.26e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 15/8·4-s + 2.35·11-s + 2.51·16-s + 0.845·19-s + 2.04·29-s + 0.486·31-s + 1.54·41-s + 4.41·44-s + 1.89·49-s − 1.23·59-s − 2.17·61-s + 2.84·64-s − 1.37·71-s + 1.58·76-s − 1.45·79-s − 2.25·89-s − 2.56·101-s − 1.88·109-s + 3.84·116-s + 2.16·121-s + 0.912·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.671273321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.671273321\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 43 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3610 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1545 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 35 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1910 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 203 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 150550 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 169230 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291030 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 280 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 581645 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 412 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 195865 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 510 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 539845 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 945 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 272830 T^{2} + p^{6} T^{4} \) |
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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
−11.91024676504293148752800662682, −11.78095794529568528931988329931, −11.01358803702866093824695124426, −10.84989625072399154783647284807, −10.13650849674616674161009520178, −9.737666004305842109144221759153, −9.045275310979518361918703915143, −8.703886115521945247123125525885, −7.81631347999061760717321019399, −7.49758683097558234352435081383, −6.73218584991688115356403756762, −6.64904153775204110324849651449, −6.01895959926985832442616390006, −5.57983661243711358920732504523, −4.42126816620113659008484445807, −3.97636953211741518269571920036, −2.96998460118144349797316446203, −2.67550728019049928456824006338, −1.33272796765119163125405994647, −1.26783699447599961263571658716,
1.26783699447599961263571658716, 1.33272796765119163125405994647, 2.67550728019049928456824006338, 2.96998460118144349797316446203, 3.97636953211741518269571920036, 4.42126816620113659008484445807, 5.57983661243711358920732504523, 6.01895959926985832442616390006, 6.64904153775204110324849651449, 6.73218584991688115356403756762, 7.49758683097558234352435081383, 7.81631347999061760717321019399, 8.703886115521945247123125525885, 9.045275310979518361918703915143, 9.737666004305842109144221759153, 10.13650849674616674161009520178, 10.84989625072399154783647284807, 11.01358803702866093824695124426, 11.78095794529568528931988329931, 11.91024676504293148752800662682
