| L(s) = 1 | + 4-s + 2·5-s + 2·7-s − 4·11-s − 3·16-s + 4·17-s + 4·19-s + 2·20-s − 8·23-s + 3·25-s + 2·28-s + 4·29-s + 12·31-s + 4·35-s + 4·37-s + 4·41-s − 4·44-s − 8·47-s + 3·49-s + 16·53-s − 8·55-s − 4·61-s − 7·64-s − 8·67-s + 4·68-s − 20·71-s − 16·73-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s + 0.755·7-s − 1.20·11-s − 3/4·16-s + 0.970·17-s + 0.917·19-s + 0.447·20-s − 1.66·23-s + 3/5·25-s + 0.377·28-s + 0.742·29-s + 2.15·31-s + 0.676·35-s + 0.657·37-s + 0.624·41-s − 0.603·44-s − 1.16·47-s + 3/7·49-s + 2.19·53-s − 1.07·55-s − 0.512·61-s − 7/8·64-s − 0.977·67-s + 0.485·68-s − 2.37·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.134578876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134578876\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−11.91449639529286679343764226550, −11.62966398931597000531890685261, −10.88584730701861158088202701344, −10.23807077573744105458890768490, −10.21125968349440310427970103854, −9.847658785994140335906524698581, −8.978317755589890469018592777050, −8.653272542026785977915399731056, −7.931019640878645555463656269107, −7.65886263843245056889902276619, −7.23932452918618201643972522127, −6.29399383983879359322286965728, −6.06427715487538538134474581204, −5.55293151834336941372141479934, −4.68941168384031411908786502640, −4.63388650382293388787577376817, −3.40959713794980443941501495212, −2.63516891990134838988787971033, −2.21877796503030880149871066016, −1.17993286506386506043984798835,
1.17993286506386506043984798835, 2.21877796503030880149871066016, 2.63516891990134838988787971033, 3.40959713794980443941501495212, 4.63388650382293388787577376817, 4.68941168384031411908786502640, 5.55293151834336941372141479934, 6.06427715487538538134474581204, 6.29399383983879359322286965728, 7.23932452918618201643972522127, 7.65886263843245056889902276619, 7.931019640878645555463656269107, 8.653272542026785977915399731056, 8.978317755589890469018592777050, 9.847658785994140335906524698581, 10.21125968349440310427970103854, 10.23807077573744105458890768490, 10.88584730701861158088202701344, 11.62966398931597000531890685261, 11.91449639529286679343764226550
