| L(s) = 1 | + 4·11-s − 4·16-s − 2·19-s + 4·29-s + 20·41-s + 5·49-s − 16·59-s + 14·61-s − 4·71-s + 6·79-s − 24·89-s + 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s − 16·176-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 16-s − 0.458·19-s + 0.742·29-s + 3.12·41-s + 5/7·49-s − 2.08·59-s + 1.79·61-s − 0.474·71-s + 0.675·79-s − 2.54·89-s + 1.91·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s − 1.20·176-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.906556328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906556328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−10.83630340410087543182393432573, −10.27598836545613570342397460810, −9.789294554980290467328803558462, −9.366843196310772627261915545415, −8.868954916520125547467951537096, −8.868353164148140775252912716414, −8.073057881143526901930445746338, −7.66840579925064627372355348308, −7.19082556914285386204002743036, −6.57318397047354120570613554836, −6.45771888340403489608808523083, −5.80145265713077003259846302271, −5.36709057899707787057901881388, −4.46596328439663925477057940537, −4.33858013797478999113915897832, −3.83580122559366398483277840751, −2.96102269980535010953420353521, −2.46422971881175287498590247921, −1.68105794284797371622869643718, −0.77052564161401378677488137200,
0.77052564161401378677488137200, 1.68105794284797371622869643718, 2.46422971881175287498590247921, 2.96102269980535010953420353521, 3.83580122559366398483277840751, 4.33858013797478999113915897832, 4.46596328439663925477057940537, 5.36709057899707787057901881388, 5.80145265713077003259846302271, 6.45771888340403489608808523083, 6.57318397047354120570613554836, 7.19082556914285386204002743036, 7.66840579925064627372355348308, 8.073057881143526901930445746338, 8.868353164148140775252912716414, 8.868954916520125547467951537096, 9.366843196310772627261915545415, 9.789294554980290467328803558462, 10.27598836545613570342397460810, 10.83630340410087543182393432573
