L(s) = 1 | − 5-s + 7-s + 11-s + 2·29-s + 31-s − 35-s − 53-s − 55-s − 2·59-s − 73-s + 77-s − 2·79-s + 83-s − 97-s − 101-s − 2·103-s + 107-s + ⋯ |
L(s) = 1 | − 5-s + 7-s + 11-s + 2·29-s + 31-s − 35-s − 53-s − 55-s − 2·59-s − 73-s + 77-s − 2·79-s + 83-s − 97-s − 101-s − 2·103-s + 107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9796344718\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9796344718\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.50187214105928336929618947342, −9.470855538428708862754037090713, −8.445270345049538062706248023315, −8.006801411374389019947139903431, −7.02638523463142839828142211229, −6.12256169823430694142836280042, −4.75608201280582108632786925101, −4.22383856368315521107451795815, −3.00702930366042590572332476086, −1.38421733695348448716042053492,
1.38421733695348448716042053492, 3.00702930366042590572332476086, 4.22383856368315521107451795815, 4.75608201280582108632786925101, 6.12256169823430694142836280042, 7.02638523463142839828142211229, 8.006801411374389019947139903431, 8.445270345049538062706248023315, 9.470855538428708862754037090713, 10.50187214105928336929618947342