| L(s) = 1 | − 4·4-s − 84·11-s + 16·16-s + 230·19-s − 420·29-s − 386·31-s − 24·41-s + 336·44-s + 685·49-s + 1.38e3·59-s − 1.46e3·61-s − 64·64-s + 456·71-s − 920·76-s + 320·79-s − 480·89-s − 1.82e3·101-s + 3.47e3·109-s + 1.68e3·116-s + 2.63e3·121-s + 1.54e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.30·11-s + 1/4·16-s + 2.77·19-s − 2.68·29-s − 2.23·31-s − 0.0914·41-s + 1.15·44-s + 1.99·49-s + 3.04·59-s − 3.07·61-s − 1/8·64-s + 0.762·71-s − 1.38·76-s + 0.455·79-s − 0.571·89-s − 1.79·101-s + 3.04·109-s + 1.34·116-s + 1.97·121-s + 1.11·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6931667952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6931667952\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 685 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 95 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6910 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 115 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1910 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 193 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 19510 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 89845 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 36250 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 260890 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 690 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 733 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 512125 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 228 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 101810 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 930130 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1564225 T^{2} + p^{6} T^{4} \) |
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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
−10.83873681475319718413332433348, −10.42358364976021129169946794179, −10.04631242273619391487746896641, −9.402775764559550739188966706582, −9.276509670919285682082372819313, −8.739591927397558823403244219767, −8.004239565543575578434790469912, −7.52123912447999566358368013415, −7.48998783759318748165935589343, −7.03886836608338327275433017618, −5.72425587523346107784076344787, −5.71360486807984569434128470588, −5.20584491168773341484575028387, −4.93138856214640589143034796965, −3.73581182558786197460205492013, −3.65225842048490600129525126996, −2.78991282878349325069165891270, −2.20996701283792806053974829948, −1.28978331279074381337272596413, −0.28117832935911331330552939217,
0.28117832935911331330552939217, 1.28978331279074381337272596413, 2.20996701283792806053974829948, 2.78991282878349325069165891270, 3.65225842048490600129525126996, 3.73581182558786197460205492013, 4.93138856214640589143034796965, 5.20584491168773341484575028387, 5.71360486807984569434128470588, 5.72425587523346107784076344787, 7.03886836608338327275433017618, 7.48998783759318748165935589343, 7.52123912447999566358368013415, 8.004239565543575578434790469912, 8.739591927397558823403244219767, 9.276509670919285682082372819313, 9.402775764559550739188966706582, 10.04631242273619391487746896641, 10.42358364976021129169946794179, 10.83873681475319718413332433348
