| L(s) = 1 | − 4-s − 4·5-s + 12·11-s + 16-s − 16·19-s + 4·20-s + 11·25-s − 8·29-s − 2·31-s − 4·41-s − 12·44-s + 10·49-s − 48·55-s + 12·59-s − 28·61-s − 64-s − 32·71-s + 16·76-s − 4·80-s + 20·89-s + 64·95-s − 11·100-s − 4·109-s + 8·116-s + 86·121-s + 2·124-s − 24·125-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s + 3.61·11-s + 1/4·16-s − 3.67·19-s + 0.894·20-s + 11/5·25-s − 1.48·29-s − 0.359·31-s − 0.624·41-s − 1.80·44-s + 10/7·49-s − 6.47·55-s + 1.56·59-s − 3.58·61-s − 1/8·64-s − 3.79·71-s + 1.83·76-s − 0.447·80-s + 2.11·89-s + 6.56·95-s − 1.09·100-s − 0.383·109-s + 0.742·116-s + 7.81·121-s + 0.179·124-s − 2.14·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6044090855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6044090855\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.174385658886192060738267710896, −8.541929243503319816189804479996, −8.507011045644721387098044728637, −7.896083085709122792290005318772, −7.46742238702381036867177553708, −7.05586869779499667883133627318, −6.70898617062002831849036029098, −6.49580116874956016634666786876, −5.93418255541737317983063657958, −5.79309548456594990933325815517, −4.72863069580394294238342544199, −4.37751231813317076756371808500, −4.14672469426042080520212108669, −4.13988336851074189264126987040, −3.38857834604733503745634490375, −3.35605328859034433015029230015, −2.21803526142860722166640311296, −1.71733769180083917199058979112, −1.19153188991427960478068159100, −0.27882036845845731905840114920,
0.27882036845845731905840114920, 1.19153188991427960478068159100, 1.71733769180083917199058979112, 2.21803526142860722166640311296, 3.35605328859034433015029230015, 3.38857834604733503745634490375, 4.13988336851074189264126987040, 4.14672469426042080520212108669, 4.37751231813317076756371808500, 4.72863069580394294238342544199, 5.79309548456594990933325815517, 5.93418255541737317983063657958, 6.49580116874956016634666786876, 6.70898617062002831849036029098, 7.05586869779499667883133627318, 7.46742238702381036867177553708, 7.896083085709122792290005318772, 8.507011045644721387098044728637, 8.541929243503319816189804479996, 9.174385658886192060738267710896
