L(s) = 1 | − 2·3-s + 3·9-s + 4·13-s + 12·17-s − 2·23-s + 25-s − 4·27-s + 2·29-s − 8·39-s − 22·43-s − 49-s − 24·51-s − 6·53-s − 8·61-s + 4·69-s − 2·75-s + 30·79-s + 5·81-s − 4·87-s − 32·101-s + 24·103-s − 8·107-s + 22·113-s + 12·117-s + 6·121-s + 127-s + 44·129-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 1.10·13-s + 2.91·17-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.28·39-s − 3.35·43-s − 1/7·49-s − 3.36·51-s − 0.824·53-s − 1.02·61-s + 0.481·69-s − 0.230·75-s + 3.37·79-s + 5/9·81-s − 0.428·87-s − 3.18·101-s + 2.36·103-s − 0.773·107-s + 2.06·113-s + 1.10·117-s + 6/11·121-s + 0.0887·127-s + 3.87·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.918129879\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.918129879\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.382727049596025346340562136311, −8.045705607407691659436078627894, −7.84537150559380198327466438679, −7.61416510260865972408388444784, −6.81892355778058308698077354679, −6.76657056581834794571008738836, −6.28699067582588991221216093683, −6.03633762594360331831201518923, −5.48660355724880677981472902540, −5.37658986480747634154607529367, −4.91921661782693105269749600413, −4.61737260061205569487576201501, −3.90650036721451309976714546461, −3.61505582217247546456430387106, −3.22335136541662494513164020154, −2.92446947994951148400058396319, −1.88597987254858056082741587211, −1.56888561835828550203111598133, −1.06947932463898115256300323382, −0.49330282133277651099906127829,
0.49330282133277651099906127829, 1.06947932463898115256300323382, 1.56888561835828550203111598133, 1.88597987254858056082741587211, 2.92446947994951148400058396319, 3.22335136541662494513164020154, 3.61505582217247546456430387106, 3.90650036721451309976714546461, 4.61737260061205569487576201501, 4.91921661782693105269749600413, 5.37658986480747634154607529367, 5.48660355724880677981472902540, 6.03633762594360331831201518923, 6.28699067582588991221216093683, 6.76657056581834794571008738836, 6.81892355778058308698077354679, 7.61416510260865972408388444784, 7.84537150559380198327466438679, 8.045705607407691659436078627894, 8.382727049596025346340562136311