| L(s) = 1 | + 4·2-s + 12·4-s − 10·5-s + 16·7-s + 32·8-s − 40·10-s − 84·11-s + 4·13-s + 64·14-s + 80·16-s − 48·17-s + 28·19-s − 120·20-s − 336·22-s − 24·23-s + 75·25-s + 16·26-s + 192·28-s − 438·29-s − 164·31-s + 192·32-s − 192·34-s − 160·35-s − 152·37-s + 112·38-s − 320·40-s − 42·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.863·7-s + 1.41·8-s − 1.26·10-s − 2.30·11-s + 0.0853·13-s + 1.22·14-s + 5/4·16-s − 0.684·17-s + 0.338·19-s − 1.34·20-s − 3.25·22-s − 0.217·23-s + 3/5·25-s + 0.120·26-s + 1.29·28-s − 2.80·29-s − 0.950·31-s + 1.06·32-s − 0.968·34-s − 0.772·35-s − 0.675·37-s + 0.478·38-s − 1.26·40-s − 0.159·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 16 T + 615 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 84 T + 3886 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T - 462 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 48 T + 326 p T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 24 T + 24343 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 438 T + 96199 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 164 T + 57666 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 152 T + 15822 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 42 T + 43 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 224 T + 80298 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 648 T + 282247 T^{2} + 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 300 T + 285694 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 672 T + 479914 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 166 T + 123351 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 76 T + 504555 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 228 T + 490678 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 464 T + 570498 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1136 T + 1070562 T^{2} + 1136 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 924 T + 471283 T^{2} - 924 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2022 T + 2378059 T^{2} + 2022 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 28 T + 250902 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.500853926785730290868786408412, −9.499444683297549434606270306099, −8.364251300304347094239105993619, −8.330294873023430569662083598282, −7.73512723599198383147382751479, −7.60952649830974471361647776470, −6.93419199349081874562480477377, −6.76296193422415471957638192144, −5.66161321133793570610866215689, −5.65059839182458998672325325883, −5.09098951688564313336875076620, −4.80594105229797667390011270263, −4.19523402124251141814428870974, −3.74093514086367974770635289770, −3.12476625793234380856791788874, −2.77901364072186355878933254709, −1.83031577113559717820424953433, −1.69906534093842282600865915387, 0, 0,
1.69906534093842282600865915387, 1.83031577113559717820424953433, 2.77901364072186355878933254709, 3.12476625793234380856791788874, 3.74093514086367974770635289770, 4.19523402124251141814428870974, 4.80594105229797667390011270263, 5.09098951688564313336875076620, 5.65059839182458998672325325883, 5.66161321133793570610866215689, 6.76296193422415471957638192144, 6.93419199349081874562480477377, 7.60952649830974471361647776470, 7.73512723599198383147382751479, 8.330294873023430569662083598282, 8.364251300304347094239105993619, 9.499444683297549434606270306099, 9.500853926785730290868786408412
