| L(s) = 1 | − 2·4-s + 8·7-s + 4·16-s + 32·19-s + 32·25-s − 16·28-s − 88·31-s + 68·37-s − 80·43-s − 50·49-s + 100·61-s − 8·64-s − 16·67-s + 32·73-s − 64·76-s − 152·79-s − 352·97-s − 64·100-s − 56·103-s + 32·112-s − 46·121-s + 176·124-s + 127-s + 131-s + 256·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 8/7·7-s + 1/4·16-s + 1.68·19-s + 1.27·25-s − 4/7·28-s − 2.83·31-s + 1.83·37-s − 1.86·43-s − 1.02·49-s + 1.63·61-s − 1/8·64-s − 0.238·67-s + 0.438·73-s − 0.842·76-s − 1.92·79-s − 3.62·97-s − 0.639·100-s − 0.543·103-s + 2/7·112-s − 0.380·121-s + 1.41·124-s + 0.00787·127-s + 0.00763·131-s + 1.92·133-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.147129800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147129800\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1664 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1184 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4160 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15680 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 176 T + p^{2} T^{2} )^{2} \) |
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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
−8.655877973957069158234667438734, −8.333854657309419154767494554444, −8.081872566620792985039413970172, −7.57889218544858530045250790087, −7.36161706514356180912487998145, −6.76994497376395916212493250034, −6.74645447179731048227561601598, −5.82569583678029344650439263092, −5.61892372421315031302368962231, −5.15743867408645896480419395751, −5.05851951870939220723576237073, −4.43848908993532829481211448703, −4.13300312628013254945760087266, −3.54356035331111375810540903062, −3.16741277860025713636543376590, −2.72562336290403614964988796548, −2.01046868097904944706702883710, −1.44146385885567393521813850292, −1.18900354804286435204380541119, −0.35141857313803264968523603261,
0.35141857313803264968523603261, 1.18900354804286435204380541119, 1.44146385885567393521813850292, 2.01046868097904944706702883710, 2.72562336290403614964988796548, 3.16741277860025713636543376590, 3.54356035331111375810540903062, 4.13300312628013254945760087266, 4.43848908993532829481211448703, 5.05851951870939220723576237073, 5.15743867408645896480419395751, 5.61892372421315031302368962231, 5.82569583678029344650439263092, 6.74645447179731048227561601598, 6.76994497376395916212493250034, 7.36161706514356180912487998145, 7.57889218544858530045250790087, 8.081872566620792985039413970172, 8.333854657309419154767494554444, 8.655877973957069158234667438734
