| L(s) = 1 | − 2-s − 5·4-s + 14·7-s + 3·8-s + 56·11-s + 52·13-s − 14·14-s − 29·16-s + 103·17-s − 57·19-s − 56·22-s + 31·23-s − 52·26-s − 70·28-s + 413·29-s − 162·31-s + 115·32-s − 103·34-s + 75·37-s + 57·38-s + 505·41-s − 73·43-s − 280·44-s − 31·46-s − 224·47-s + 147·49-s − 260·52-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 5/8·4-s + 0.755·7-s + 0.132·8-s + 1.53·11-s + 1.10·13-s − 0.267·14-s − 0.453·16-s + 1.46·17-s − 0.688·19-s − 0.542·22-s + 0.281·23-s − 0.392·26-s − 0.472·28-s + 2.64·29-s − 0.938·31-s + 0.635·32-s − 0.519·34-s + 0.333·37-s + 0.243·38-s + 1.92·41-s − 0.258·43-s − 0.959·44-s − 0.0993·46-s − 0.695·47-s + 3/7·49-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.511041469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.511041469\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 56 T + 2421 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 p T + 4414 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 103 T + 12222 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 p T + 14028 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 31 T + 24318 T^{2} - 31 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 413 T + 83948 T^{2} - 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 162 T + 21494 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 75 T + 69410 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 505 T + 163458 T^{2} - 505 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 73 T + 99574 T^{2} + 73 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 224 T + 100634 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 262 T + 239106 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 190 T - 32242 T^{2} + 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 990 T + 669098 T^{2} - 990 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 908 T + 525193 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 127 T - 221188 T^{2} + 127 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 337 T + 64726 T^{2} + 337 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1119 T + 1281888 T^{2} + 1119 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1517 T + 1670096 T^{2} - 1517 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1713 T + 2140568 T^{2} - 1713 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1764 T + 1844934 T^{2} - 1764 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.156103236349722025099759684744, −8.838134311232744556932940896538, −8.447960317874745088348632089854, −8.325848433073911748200070429121, −7.76223597770825459068182732620, −7.35710330026062763203954688928, −6.78598713248978794164769665766, −6.37232216039071550683442061272, −6.17398822645377189219624232740, −5.60939127890603190249419851742, −4.97148196796488684011996480558, −4.78176142832046828881042031172, −4.06921069039322911005732396761, −3.91705952944725557138387998311, −3.39294232959851752633017615218, −2.70279145089734958656199450452, −2.10436174858164533486591129343, −1.40869747973902791101569983002, −0.818406656170659476356281396700, −0.75172354341462870711658817546,
0.75172354341462870711658817546, 0.818406656170659476356281396700, 1.40869747973902791101569983002, 2.10436174858164533486591129343, 2.70279145089734958656199450452, 3.39294232959851752633017615218, 3.91705952944725557138387998311, 4.06921069039322911005732396761, 4.78176142832046828881042031172, 4.97148196796488684011996480558, 5.60939127890603190249419851742, 6.17398822645377189219624232740, 6.37232216039071550683442061272, 6.78598713248978794164769665766, 7.35710330026062763203954688928, 7.76223597770825459068182732620, 8.325848433073911748200070429121, 8.447960317874745088348632089854, 8.838134311232744556932940896538, 9.156103236349722025099759684744
