| L(s) = 1 | − 2·2-s + 2·5-s − 12·7-s + 8·8-s − 4·10-s + 12·11-s − 2·13-s + 24·14-s − 16·16-s + 20·17-s − 24·22-s + 48·23-s + 25·25-s + 4·26-s + 26·29-s − 12·31-s − 40·34-s − 24·35-s + 52·37-s + 16·40-s − 58·41-s − 84·43-s − 96·46-s + 120·47-s + 47·49-s − 50·50-s − 148·53-s + ⋯ |
| L(s) = 1 | − 2-s + 2/5·5-s − 1.71·7-s + 8-s − 2/5·10-s + 1.09·11-s − 0.153·13-s + 12/7·14-s − 16-s + 1.17·17-s − 1.09·22-s + 2.08·23-s + 25-s + 2/13·26-s + 0.896·29-s − 0.387·31-s − 1.17·34-s − 0.685·35-s + 1.40·37-s + 2/5·40-s − 1.41·41-s − 1.95·43-s − 2.08·46-s + 2.55·47-s + 0.959·49-s − 50-s − 2.79·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.132394220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132394220\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T + 169 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T - 165 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 48 T + 1297 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 26 T - 165 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T + 1009 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 58 T + 1683 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 84 T + 4201 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 120 T + 7009 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 156 T + 11593 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T - 3045 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 4537 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 204 T + 20113 T^{2} - 204 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 84 T + 9241 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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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
−11.61770078535850364858827475962, −10.96806583048774472070768393325, −10.39922169375727536674065875759, −10.21584740401443396056636177427, −9.490535468864704929459242114995, −9.488303515669845521913963377311, −8.942786680086825691965542683136, −8.584395638076591572355753045050, −7.88131312678645962044326944552, −7.30021008437716485784506308365, −6.76003593454566117469580886377, −6.53338437473268137078941147917, −5.92975165906661750374051382539, −4.98763527613806650993030162079, −4.79202279149799065230097647623, −3.57269414473983624257874887053, −3.39063690577401937976608389945, −2.50923674544035722318272662905, −1.30010437368648427485667827724, −0.68914308463710104276843717446,
0.68914308463710104276843717446, 1.30010437368648427485667827724, 2.50923674544035722318272662905, 3.39063690577401937976608389945, 3.57269414473983624257874887053, 4.79202279149799065230097647623, 4.98763527613806650993030162079, 5.92975165906661750374051382539, 6.53338437473268137078941147917, 6.76003593454566117469580886377, 7.30021008437716485784506308365, 7.88131312678645962044326944552, 8.584395638076591572355753045050, 8.942786680086825691965542683136, 9.488303515669845521913963377311, 9.490535468864704929459242114995, 10.21584740401443396056636177427, 10.39922169375727536674065875759, 10.96806583048774472070768393325, 11.61770078535850364858827475962
