L(s) = 1 | + 8·7-s − 9-s − 12·17-s + 6·25-s + 8·31-s − 4·41-s + 16·47-s + 34·49-s − 8·63-s + 32·71-s + 12·73-s + 8·79-s + 81-s − 20·89-s − 28·97-s + 24·103-s + 4·113-s − 96·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + ⋯ |
L(s) = 1 | + 3.02·7-s − 1/3·9-s − 2.91·17-s + 6/5·25-s + 1.43·31-s − 0.624·41-s + 2.33·47-s + 34/7·49-s − 1.00·63-s + 3.79·71-s + 1.40·73-s + 0.900·79-s + 1/9·81-s − 2.11·89-s − 2.84·97-s + 2.36·103-s + 0.376·113-s − 8.80·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.744450027\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.744450027\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−10.67946436509313668025194157594, −10.37942400124764987371244932936, −9.593903618034796381942119017535, −9.038429883758629860250245227080, −8.628872410757519339559271625673, −8.545381017011757937158500388896, −7.999934091137468204418698566077, −7.73787552333724693574473136306, −7.02870266290699155267383067121, −6.67164596864510079035274307057, −6.25607939173399545262595113262, −5.36826190608927672454462656452, −4.93707917873473456252512421277, −4.90976972841541250032652808862, −4.14469470417919856951895077712, −3.97097804215351266167771249436, −2.48635200768358468656017896673, −2.46312818680148202985242208834, −1.69566073913797812775921081438, −0.898833149425865420910600908462,
0.898833149425865420910600908462, 1.69566073913797812775921081438, 2.46312818680148202985242208834, 2.48635200768358468656017896673, 3.97097804215351266167771249436, 4.14469470417919856951895077712, 4.90976972841541250032652808862, 4.93707917873473456252512421277, 5.36826190608927672454462656452, 6.25607939173399545262595113262, 6.67164596864510079035274307057, 7.02870266290699155267383067121, 7.73787552333724693574473136306, 7.999934091137468204418698566077, 8.545381017011757937158500388896, 8.628872410757519339559271625673, 9.038429883758629860250245227080, 9.593903618034796381942119017535, 10.37942400124764987371244932936, 10.67946436509313668025194157594