| L(s) = 1 | − 2-s + 4-s − 8-s + 4·13-s + 16-s + 6·17-s − 10·25-s − 4·26-s − 12·29-s − 32-s − 6·34-s − 8·37-s − 18·41-s − 10·49-s + 10·50-s + 4·52-s − 24·53-s + 12·58-s + 16·61-s + 64-s + 6·68-s + 22·73-s + 8·74-s + 18·82-s − 12·89-s + 10·97-s + 10·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 2·25-s − 0.784·26-s − 2.22·29-s − 0.176·32-s − 1.02·34-s − 1.31·37-s − 2.81·41-s − 1.42·49-s + 1.41·50-s + 0.554·52-s − 3.29·53-s + 1.57·58-s + 2.04·61-s + 1/8·64-s + 0.727·68-s + 2.57·73-s + 0.929·74-s + 1.98·82-s − 1.27·89-s + 1.01·97-s + 1.01·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−8.789487803962222669402730529275, −8.255589340941165804909076040404, −7.80670854475898716623947697687, −7.79453288119723697641112445352, −6.69953527334383655382904548898, −6.68619713403218599725876916726, −5.91234365712937538559209143490, −5.34383976963704470462884839275, −5.13777464868847606985261506723, −3.86098888246252906299834258338, −3.65076060485342096132488694617, −3.10746305536689930220572132603, −1.80773362483917328069437252787, −1.60693251106342657124988453680, 0,
1.60693251106342657124988453680, 1.80773362483917328069437252787, 3.10746305536689930220572132603, 3.65076060485342096132488694617, 3.86098888246252906299834258338, 5.13777464868847606985261506723, 5.34383976963704470462884839275, 5.91234365712937538559209143490, 6.68619713403218599725876916726, 6.69953527334383655382904548898, 7.79453288119723697641112445352, 7.80670854475898716623947697687, 8.255589340941165804909076040404, 8.789487803962222669402730529275
