| L(s) = 1 | + 8·7-s + 2·9-s − 88·23-s − 25·25-s − 124·41-s − 8·47-s − 50·49-s + 16·63-s − 77·81-s + 284·89-s − 88·103-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 704·161-s + 163-s + 167-s − 338·169-s + 173-s − 200·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 2/9·9-s − 3.82·23-s − 25-s − 3.02·41-s − 0.170·47-s − 1.02·49-s + 0.253·63-s − 0.950·81-s + 3.19·89-s − 0.854·103-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 4.37·161-s + 0.00613·163-s + 0.00598·167-s − 2·169-s + 0.00578·173-s − 8/7·175-s + 0.00558·179-s + 0.00552·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9623599343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9623599343\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1198 T^{2} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2078 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 4078 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4478 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8002 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−10.03612846821023013405151884770, −9.498152808843381694487202196570, −8.642956586599492319460931238390, −8.347552801404107060768424366550, −8.143615918766972969546680032245, −7.70591899391483880854733503361, −7.46229491018060937391155755195, −6.57110525948104310809410478123, −6.54652813471678890876191767522, −5.75599298268049254974872578162, −5.68010376112054856548518757064, −4.82103890431018628703882512346, −4.72815769083773613295609345192, −3.98323644382085895320715536571, −3.73635708464003124287213319690, −3.14318682407694803291933234169, −2.11309608291043535060006742634, −1.93348279643918723006202313487, −1.48696515690453846194386539296, −0.27021453911285295486928429482,
0.27021453911285295486928429482, 1.48696515690453846194386539296, 1.93348279643918723006202313487, 2.11309608291043535060006742634, 3.14318682407694803291933234169, 3.73635708464003124287213319690, 3.98323644382085895320715536571, 4.72815769083773613295609345192, 4.82103890431018628703882512346, 5.68010376112054856548518757064, 5.75599298268049254974872578162, 6.54652813471678890876191767522, 6.57110525948104310809410478123, 7.46229491018060937391155755195, 7.70591899391483880854733503361, 8.143615918766972969546680032245, 8.347552801404107060768424366550, 8.642956586599492319460931238390, 9.498152808843381694487202196570, 10.03612846821023013405151884770
