L(s) = 1 | − 5-s + 17-s + 19-s − 23-s + 25-s + 31-s + 2·47-s + 49-s + 53-s − 61-s + 79-s − 83-s − 85-s − 95-s + 2·107-s − 109-s − 2·113-s + 115-s + ⋯ |
L(s) = 1 | − 5-s + 17-s + 19-s − 23-s + 25-s + 31-s + 2·47-s + 49-s + 53-s − 61-s + 79-s − 83-s − 85-s − 95-s + 2·107-s − 109-s − 2·113-s + 115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.013488584\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.013488584\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 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
−9.197501538083556139194048344511, −8.383012959599458612866921371869, −7.67418889776908494798087972900, −7.18084459170572762984781401077, −6.07882265103692941486810612706, −5.28267855553206133153926001643, −4.28525604664020836258005472681, −3.56562619834789970709176886978, −2.60475014698954556892414688079, −1.01831817351789468000219260069,
1.01831817351789468000219260069, 2.60475014698954556892414688079, 3.56562619834789970709176886978, 4.28525604664020836258005472681, 5.28267855553206133153926001643, 6.07882265103692941486810612706, 7.18084459170572762984781401077, 7.67418889776908494798087972900, 8.383012959599458612866921371869, 9.197501538083556139194048344511