| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 8·10-s + 11-s − 6·12-s − 4·14-s + 8·15-s + 5·16-s + 10·17-s + 6·18-s − 5·19-s − 12·20-s + 4·21-s + 2·22-s − 5·23-s − 8·24-s + 2·25-s − 4·27-s − 6·28-s + 5·29-s + 16·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 2.52·10-s + 0.301·11-s − 1.73·12-s − 1.06·14-s + 2.06·15-s + 5/4·16-s + 2.42·17-s + 1.41·18-s − 1.14·19-s − 2.68·20-s + 0.872·21-s + 0.426·22-s − 1.04·23-s − 1.63·24-s + 2/5·25-s − 0.769·27-s − 1.13·28-s + 0.928·29-s + 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.048710514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.048710514\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 128 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 19 T + 192 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 122 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 182 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 134 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.71106959088840753550387500548, −7.69871111019400762817742605765, −7.43509160361300888961752590891, −7.08816454063471990109089952084, −6.33232502687775390913158752696, −6.27652365749271618406848516170, −6.05709995909393316967213813713, −5.80249548551148513481702000736, −5.10860354500946850157457890702, −4.98076320854394265318168528350, −4.42709539493658177165003774894, −4.18141025143720980866938991078, −3.86380836381856699316487755607, −3.57319143217212327214985573362, −3.00782110969579809836318808340, −3.00729679835614347173655415340, −1.92093051885760494544825415826, −1.74975107276097470707868923911, −0.73323168133143140344665925127, −0.52027437920800937807293772610,
0.52027437920800937807293772610, 0.73323168133143140344665925127, 1.74975107276097470707868923911, 1.92093051885760494544825415826, 3.00729679835614347173655415340, 3.00782110969579809836318808340, 3.57319143217212327214985573362, 3.86380836381856699316487755607, 4.18141025143720980866938991078, 4.42709539493658177165003774894, 4.98076320854394265318168528350, 5.10860354500946850157457890702, 5.80249548551148513481702000736, 6.05709995909393316967213813713, 6.27652365749271618406848516170, 6.33232502687775390913158752696, 7.08816454063471990109089952084, 7.43509160361300888961752590891, 7.69871111019400762817742605765, 7.71106959088840753550387500548
