| L(s) = 1 | − 8·4-s − 146·7-s + 190·13-s + 64·16-s − 626·19-s + 98·25-s + 1.16e3·28-s − 1.91e3·31-s − 770·37-s + 5.09e3·43-s + 1.11e4·49-s − 1.52e3·52-s + 1.12e4·61-s − 512·64-s + 46·67-s + 1.30e4·73-s + 5.00e3·76-s − 1.22e4·79-s − 2.77e4·91-s + 1.98e4·97-s − 784·100-s + 1.24e4·103-s − 3.19e4·109-s − 9.34e3·112-s + 482·121-s + 1.53e4·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.97·7-s + 1.12·13-s + 1/4·16-s − 1.73·19-s + 0.156·25-s + 1.48·28-s − 1.99·31-s − 0.562·37-s + 2.75·43-s + 4.65·49-s − 0.562·52-s + 3.01·61-s − 1/8·64-s + 0.0102·67-s + 2.44·73-s + 0.867·76-s − 1.96·79-s − 3.34·91-s + 2.11·97-s − 0.0783·100-s + 1.17·103-s − 2.69·109-s − 0.744·112-s + 0.0329·121-s + 0.996·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6597983119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6597983119\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 98 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 73 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 482 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 95 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 156674 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 313 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 143810 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 1023070 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 958 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 385 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6722 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2546 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9730562 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8772194 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 17045090 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5615 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 23 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50657474 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6527 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6121 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 88608290 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 117926210 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9935 T + p^{4} T^{2} )^{2} \) |
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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
−14.85946600044083147208533675743, −14.27929837325142025237977421379, −13.55034290483143455189192796405, −13.00715109311022080394828296901, −12.62673029360125138779351373876, −12.61818407764360007899384369307, −11.32259220085438747236671740762, −10.64909231794088292713826833054, −10.17164481793123421347491472856, −9.457065091914922316500266648783, −9.048139527762484432975986594731, −8.534458999813907815790966575366, −7.35971710416587560199652493280, −6.60473207587299040343893441047, −6.20629506934499004116996626786, −5.49800087083757242443842709417, −3.85814826320137219916880765952, −3.72597869479421022159456482301, −2.51718980338395994914988837242, −0.46999768446800538760205809946,
0.46999768446800538760205809946, 2.51718980338395994914988837242, 3.72597869479421022159456482301, 3.85814826320137219916880765952, 5.49800087083757242443842709417, 6.20629506934499004116996626786, 6.60473207587299040343893441047, 7.35971710416587560199652493280, 8.534458999813907815790966575366, 9.048139527762484432975986594731, 9.457065091914922316500266648783, 10.17164481793123421347491472856, 10.64909231794088292713826833054, 11.32259220085438747236671740762, 12.61818407764360007899384369307, 12.62673029360125138779351373876, 13.00715109311022080394828296901, 13.55034290483143455189192796405, 14.27929837325142025237977421379, 14.85946600044083147208533675743
