L(s) = 1 | − 4·3-s + 4-s + 6·9-s − 4·12-s + 2·13-s + 16-s − 2·17-s − 10·25-s + 4·27-s + 6·36-s − 8·39-s + 16·43-s − 4·48-s + 2·49-s + 8·51-s + 2·52-s − 12·53-s − 8·61-s + 64-s − 2·68-s + 40·75-s + 16·79-s − 37·81-s − 10·100-s + 36·101-s − 32·103-s − 12·107-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 2·25-s + 0.769·27-s + 36-s − 1.28·39-s + 2.43·43-s − 0.577·48-s + 2/7·49-s + 1.12·51-s + 0.277·52-s − 1.64·53-s − 1.02·61-s + 1/8·64-s − 0.242·68-s + 4.61·75-s + 1.80·79-s − 4.11·81-s − 100-s + 3.58·101-s − 3.15·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.130451766940036633976492592874, −8.178809137350784472556578192263, −7.87934314273660946672510921120, −7.27327240971263335966374947457, −6.65330731261113455462227374582, −6.23937891378160525062636317543, −5.99911855171913780844071238816, −5.55291225676643931119698422120, −5.07854511056822008543327271928, −4.41754598098690674415639630310, −3.90229547122613769308947697962, −2.98027300843893188865212770683, −2.11773918281950756189912402247, −1.08527840922908392734081376863, 0,
1.08527840922908392734081376863, 2.11773918281950756189912402247, 2.98027300843893188865212770683, 3.90229547122613769308947697962, 4.41754598098690674415639630310, 5.07854511056822008543327271928, 5.55291225676643931119698422120, 5.99911855171913780844071238816, 6.23937891378160525062636317543, 6.65330731261113455462227374582, 7.27327240971263335966374947457, 7.87934314273660946672510921120, 8.178809137350784472556578192263, 9.130451766940036633976492592874