L(s) = 1 | + 2·5-s + 9-s − 11-s − 17-s + 19-s − 23-s + 3·25-s + 2·43-s + 2·45-s − 47-s − 2·55-s − 61-s − 73-s + 81-s − 83-s − 2·85-s + 2·95-s − 99-s − 101-s − 2·115-s + ⋯ |
L(s) = 1 | + 2·5-s + 9-s − 11-s − 17-s + 19-s − 23-s + 3·25-s + 2·43-s + 2·45-s − 47-s − 2·55-s − 61-s − 73-s + 81-s − 83-s − 2·85-s + 2·95-s − 99-s − 101-s − 2·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.900694499\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.900694499\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.954664823668009630851556990411, −7.899163186661949798659791371955, −7.13560091110797130603473061307, −6.37509002168219643461650163128, −5.73034007257154187992722417210, −5.06583715507356832249083395124, −4.27596179986279209919326319998, −2.89881406933666699592433389080, −2.18160689766123360288191491521, −1.36074522020081191294207388730,
1.36074522020081191294207388730, 2.18160689766123360288191491521, 2.89881406933666699592433389080, 4.27596179986279209919326319998, 5.06583715507356832249083395124, 5.73034007257154187992722417210, 6.37509002168219643461650163128, 7.13560091110797130603473061307, 7.899163186661949798659791371955, 8.954664823668009630851556990411