| L(s) = 1 | + 6·5-s + 14·7-s + 44·11-s + 14·13-s − 96·17-s − 170·19-s + 152·23-s − 186·25-s + 128·29-s − 68·31-s + 84·35-s − 256·37-s − 88·41-s − 724·43-s − 244·47-s + 147·49-s − 188·53-s + 264·55-s + 138·59-s − 358·61-s + 84·65-s − 200·67-s − 1.40e3·71-s − 628·73-s + 616·77-s − 200·79-s + 618·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + 0.755·7-s + 1.20·11-s + 0.298·13-s − 1.36·17-s − 2.05·19-s + 1.37·23-s − 1.48·25-s + 0.819·29-s − 0.393·31-s + 0.405·35-s − 1.13·37-s − 0.335·41-s − 2.56·43-s − 0.757·47-s + 3/7·49-s − 0.487·53-s + 0.647·55-s + 0.304·59-s − 0.751·61-s + 0.160·65-s − 0.364·67-s − 2.34·71-s − 1.00·73-s + 0.911·77-s − 0.284·79-s + 0.817·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T + 222 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 p T + 2998 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 14 T + 4406 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 96 T + 8430 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 170 T + 17946 T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 152 T + 20638 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 128 T + 10102 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 68 T + 35726 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 256 T + 117542 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 88 T + 15310 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 724 T + 278070 T^{2} + 724 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 244 T + 204622 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 188 T + 155038 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 138 T + 375226 T^{2} - 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 358 T + 483006 T^{2} + 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 200 T - 155706 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1400 T + 1176814 T^{2} + 1400 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 628 T + 378758 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 200 T + 926 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 618 T + 303658 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 908 T + 1379254 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1520 T + 2349518 T^{2} - 1520 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.655643819433202468799181101452, −8.523381416172241893455819510551, −7.66226553290453780254472267740, −7.60426010800305245544545303624, −6.73995742008862499437080114532, −6.69954373434263191713489086720, −6.23471316195398466899863193226, −6.06044877329638975958082847661, −5.28703153071182399397448182599, −4.89823361379572025788599178185, −4.50807887938270829097540419485, −4.26077798812884394797989861269, −3.47499191965696415797952423768, −3.36337467745996097644634163361, −2.37208942969936049019054949669, −2.06609802509812408309888621105, −1.55281134773876352362106593404, −1.25807058524034808628801635659, 0, 0,
1.25807058524034808628801635659, 1.55281134773876352362106593404, 2.06609802509812408309888621105, 2.37208942969936049019054949669, 3.36337467745996097644634163361, 3.47499191965696415797952423768, 4.26077798812884394797989861269, 4.50807887938270829097540419485, 4.89823361379572025788599178185, 5.28703153071182399397448182599, 6.06044877329638975958082847661, 6.23471316195398466899863193226, 6.69954373434263191713489086720, 6.73995742008862499437080114532, 7.60426010800305245544545303624, 7.66226553290453780254472267740, 8.523381416172241893455819510551, 8.655643819433202468799181101452
