| L(s) = 1 | + 3·2-s + 8·4-s + 5·5-s − 28·7-s + 45·8-s + 15·10-s − 45·11-s + 118·13-s − 84·14-s + 135·16-s − 54·17-s + 121·19-s + 40·20-s − 135·22-s + 69·23-s + 354·26-s − 224·28-s + 324·29-s + 88·31-s + 360·32-s − 162·34-s − 140·35-s + 259·37-s + 363·38-s + 225·40-s − 390·41-s − 572·43-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 4-s + 0.447·5-s − 1.51·7-s + 1.98·8-s + 0.474·10-s − 1.23·11-s + 2.51·13-s − 1.60·14-s + 2.10·16-s − 0.770·17-s + 1.46·19-s + 0.447·20-s − 1.30·22-s + 0.625·23-s + 2.67·26-s − 1.51·28-s + 2.07·29-s + 0.509·31-s + 1.98·32-s − 0.817·34-s − 0.676·35-s + 1.15·37-s + 1.54·38-s + 0.889·40-s − 1.48·41-s − 2.02·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.580411870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.580411870\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 45 T + 694 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 59 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 54 T - 1997 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 121 T + 7782 T^{2} - 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 p T - 14 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 88 T - 22047 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 p T + 12 p^{2} T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 195 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 286 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T - 101798 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 597 T + 207532 T^{2} - 597 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 360 T - 75779 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 392 T - 73317 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 280 T - 222363 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 668 T + 57207 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 782 T + 118485 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1194 T + 720667 T^{2} + 1194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 902 T + p^{3} T^{2} )^{2} \) |
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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
−11.45344480480780197432930795485, −11.15339304792633964524612685775, −10.30637304654184457781532351652, −10.24593016683928950709509079292, −10.01902761669125130777651336784, −9.090601050003599218191008792564, −8.382529561121609237670508893506, −8.326285987154752918201541311503, −7.37064360620068434728858112806, −6.93323746119566942113684055361, −6.54002760955191148778777047398, −5.81579668198679918505871562914, −5.78033631631063722413968915196, −4.72263679836319394521660015244, −4.55306617318921454797785601829, −3.41299952749232803619568782947, −3.30621967978097006843078618470, −2.59826281106166696326709631854, −1.54048225601424233423591348144, −0.847173441393604463168246249048,
0.847173441393604463168246249048, 1.54048225601424233423591348144, 2.59826281106166696326709631854, 3.30621967978097006843078618470, 3.41299952749232803619568782947, 4.55306617318921454797785601829, 4.72263679836319394521660015244, 5.78033631631063722413968915196, 5.81579668198679918505871562914, 6.54002760955191148778777047398, 6.93323746119566942113684055361, 7.37064360620068434728858112806, 8.326285987154752918201541311503, 8.382529561121609237670508893506, 9.090601050003599218191008792564, 10.01902761669125130777651336784, 10.24593016683928950709509079292, 10.30637304654184457781532351652, 11.15339304792633964524612685775, 11.45344480480780197432930795485
