| L(s) = 1 | − 3·5-s − 7-s − 3·11-s + 4·13-s + 3·17-s − 2·19-s − 6·23-s + 5·25-s + 6·29-s + 2·31-s + 3·35-s − 2·37-s − 43-s − 21·47-s − 5·49-s − 6·53-s + 9·55-s − 11·61-s − 12·65-s − 4·67-s − 24·71-s − 5·73-s + 3·77-s − 16·79-s + 6·83-s − 9·85-s − 18·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 0.904·11-s + 1.10·13-s + 0.727·17-s − 0.458·19-s − 1.25·23-s + 25-s + 1.11·29-s + 0.359·31-s + 0.507·35-s − 0.328·37-s − 0.152·43-s − 3.06·47-s − 5/7·49-s − 0.824·53-s + 1.21·55-s − 1.40·61-s − 1.48·65-s − 0.488·67-s − 2.84·71-s − 0.585·73-s + 0.341·77-s − 1.80·79-s + 0.658·83-s − 0.976·85-s − 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + T + 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 21 T + 196 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 144 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 144 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.491889982084046144249139008598, −8.423698665575543106001201384415, −7.75637007124788722904540893489, −7.55493470393042401050202205629, −7.44671121317034853400265251667, −6.54923593566407051824565505092, −6.32565818442085182321759233482, −6.16173934535562558501669190666, −5.57873378733363067590571790105, −4.94167707585443375706508828860, −4.68101691325403769622023861025, −4.33755312834980223530662977223, −3.61874723484322138287259115311, −3.52307376353046495741585711033, −2.94408676295673668246561280874, −2.63598484346465082372558884813, −1.59639676538250773195616108165, −1.32173154835776330094994971518, 0, 0,
1.32173154835776330094994971518, 1.59639676538250773195616108165, 2.63598484346465082372558884813, 2.94408676295673668246561280874, 3.52307376353046495741585711033, 3.61874723484322138287259115311, 4.33755312834980223530662977223, 4.68101691325403769622023861025, 4.94167707585443375706508828860, 5.57873378733363067590571790105, 6.16173934535562558501669190666, 6.32565818442085182321759233482, 6.54923593566407051824565505092, 7.44671121317034853400265251667, 7.55493470393042401050202205629, 7.75637007124788722904540893489, 8.423698665575543106001201384415, 8.491889982084046144249139008598
