L(s) = 1 | + 3-s − 5-s + 7-s + 13-s − 15-s − 17-s + 21-s − 27-s − 2·31-s − 35-s − 37-s + 39-s + 43-s + 47-s − 51-s − 65-s + 71-s − 81-s + 85-s + 91-s − 2·93-s − 105-s − 2·107-s − 109-s − 111-s + 2·113-s − 119-s + ⋯ |
L(s) = 1 | + 3-s − 5-s + 7-s + 13-s − 15-s − 17-s + 21-s − 27-s − 2·31-s − 35-s − 37-s + 39-s + 43-s + 47-s − 51-s − 65-s + 71-s − 81-s + 85-s + 91-s − 2·93-s − 105-s − 2·107-s − 109-s − 111-s + 2·113-s − 119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.024488922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.024488922\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−11.21920871375829792749765569162, −10.88130085406947463401502402381, −9.250450978319346720755416548801, −8.585287277419569846017455910415, −7.954143502560078217961115903988, −7.10955820197510546801346318054, −5.61729941197096691827675041220, −4.26984902056982014947676898937, −3.47895634233078360386963741748, −1.99755916622639326258291511234,
1.99755916622639326258291511234, 3.47895634233078360386963741748, 4.26984902056982014947676898937, 5.61729941197096691827675041220, 7.10955820197510546801346318054, 7.954143502560078217961115903988, 8.585287277419569846017455910415, 9.250450978319346720755416548801, 10.88130085406947463401502402381, 11.21920871375829792749765569162