| L(s) = 1 | + 60·4-s + 2.18e3·16-s − 848·25-s − 3.00e3·37-s + 1.12e4·43-s + 6.19e4·64-s − 1.30e4·67-s − 3.46e4·79-s − 5.08e4·100-s − 3.74e4·109-s + 1.36e4·121-s + 127-s + 131-s + 137-s + 139-s − 1.80e5·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.31e4·169-s + 6.76e5·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 15/4·4-s + 8.54·16-s − 1.35·25-s − 2.19·37-s + 6.10·43-s + 15.1·64-s − 2.91·67-s − 5.55·79-s − 5.08·100-s − 3.15·109-s + 0.930·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 8.21·148-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 1.86·169-s + 22.8·172-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(9.861747727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.861747727\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - 15 p T^{2} + p^{8} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 424 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6810 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 26578 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 125192 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 74158 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 548730 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 896400 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 48782 p T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 750 T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 2261672 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2820 T + p^{4} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 7080962 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 13186320 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 21556322 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 26352482 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3272 T + p^{4} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 13112070 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 12519182 T^{2} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8668 T + p^{4} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 84203042 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 85481368 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 132781262 T^{2} + p^{8} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24524861043705979131669416075, −7.23133029346364999405564494343, −7.15165147623251958543425152534, −6.65794330353688843903158551675, −6.34245343167071113208473077948, −6.13030193882598134132213899485, −6.08981074406176609312022355541, −5.66581299432505493802412933323, −5.64464514937013154969402555732, −5.42405829027767351175344501241, −4.99570131828372227738881544170, −4.38189466477954906131176013840, −4.13426251089297761996980183903, −3.94653184017170243431135554613, −3.64822700908319192508448111734, −3.16892231421853165122428830492, −2.79879220237898494257136415876, −2.74393752075831517712315156166, −2.41333894645774114218930813441, −2.34000622103695516791838434119, −1.59639431199006439923801906019, −1.48612880867300154537027360352, −1.44132485418858153170814689894, −0.864736429040300031618805394861, −0.25202749214099847738989784953,
0.25202749214099847738989784953, 0.864736429040300031618805394861, 1.44132485418858153170814689894, 1.48612880867300154537027360352, 1.59639431199006439923801906019, 2.34000622103695516791838434119, 2.41333894645774114218930813441, 2.74393752075831517712315156166, 2.79879220237898494257136415876, 3.16892231421853165122428830492, 3.64822700908319192508448111734, 3.94653184017170243431135554613, 4.13426251089297761996980183903, 4.38189466477954906131176013840, 4.99570131828372227738881544170, 5.42405829027767351175344501241, 5.64464514937013154969402555732, 5.66581299432505493802412933323, 6.08981074406176609312022355541, 6.13030193882598134132213899485, 6.34245343167071113208473077948, 6.65794330353688843903158551675, 7.15165147623251958543425152534, 7.23133029346364999405564494343, 7.24524861043705979131669416075
