L(s) = 1 | − 4·2-s − 2·3-s + 12·4-s + 10·5-s + 8·6-s − 32·8-s − 33·9-s − 40·10-s − 26·11-s − 24·12-s + 102·13-s − 20·15-s + 80·16-s − 186·17-s + 132·18-s − 36·19-s + 120·20-s + 104·22-s − 44·23-s + 64·24-s + 75·25-s − 408·26-s + 86·27-s − 46·29-s + 80·30-s − 140·31-s − 192·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.384·3-s + 3/2·4-s + 0.894·5-s + 0.544·6-s − 1.41·8-s − 1.22·9-s − 1.26·10-s − 0.712·11-s − 0.577·12-s + 2.17·13-s − 0.344·15-s + 5/4·16-s − 2.65·17-s + 1.72·18-s − 0.434·19-s + 1.34·20-s + 1.00·22-s − 0.398·23-s + 0.544·24-s + 3/5·25-s − 3.07·26-s + 0.612·27-s − 0.294·29-s + 0.486·30-s − 0.811·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $D_{4}$ | \( 1 + 2 T + 37 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 26 T + 2703 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 102 T + 6833 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 186 T + 17897 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 13530 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 44 T + 192 p T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 46 T + 38939 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 140 T + 38944 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 132 T + 28044 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 132 T + 100148 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 324 T + 181386 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 242 T + 215325 T^{2} + 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 208 T + 205512 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 780 T + 529576 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 216 T + 425298 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 256 T + 62452 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 692 T + 813066 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 832 T + 948202 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1962 T + 1889271 T^{2} + 1962 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1712 T + 1530198 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1960 T + 2217986 T^{2} + 1960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 102 T + 453465 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.27847497961099059308136024120, −10.01005595064845591322023479731, −9.144159992658927858809132564426, −8.972528334284393192014314925225, −8.539849586144574296658812875702, −8.537352906114624963476751133937, −7.64456748925158715219529819187, −7.22174967099541982022648114048, −6.41090971389846369568056675257, −6.33336961916090019116034444198, −5.67541778701140099340676040458, −5.65901961234132548134986104247, −4.46839766663131490311202300412, −4.04970000284833102842513607455, −2.80662065404681690344670275824, −2.76323900378937767482222063852, −1.79451695010414984815766840004, −1.33921706017389135457801376258, 0, 0,
1.33921706017389135457801376258, 1.79451695010414984815766840004, 2.76323900378937767482222063852, 2.80662065404681690344670275824, 4.04970000284833102842513607455, 4.46839766663131490311202300412, 5.65901961234132548134986104247, 5.67541778701140099340676040458, 6.33336961916090019116034444198, 6.41090971389846369568056675257, 7.22174967099541982022648114048, 7.64456748925158715219529819187, 8.537352906114624963476751133937, 8.539849586144574296658812875702, 8.972528334284393192014314925225, 9.144159992658927858809132564426, 10.01005595064845591322023479731, 10.27847497961099059308136024120