L(s) = 1 | + 4·5-s − 2·7-s − 5·11-s − 2·13-s − 3·17-s + 19-s − 6·23-s + 11·25-s − 2·29-s − 4·31-s − 8·35-s − 8·37-s + 41-s − 7·43-s + 2·47-s − 3·49-s − 4·53-s − 20·55-s + 5·59-s − 8·65-s − 13·67-s + 8·71-s + 3·73-s + 10·77-s + 8·79-s + 12·83-s − 12·85-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.755·7-s − 1.50·11-s − 0.554·13-s − 0.727·17-s + 0.229·19-s − 1.25·23-s + 11/5·25-s − 0.371·29-s − 0.718·31-s − 1.35·35-s − 1.31·37-s + 0.156·41-s − 1.06·43-s + 0.291·47-s − 3/7·49-s − 0.549·53-s − 2.69·55-s + 0.650·59-s − 0.992·65-s − 1.58·67-s + 0.949·71-s + 0.351·73-s + 1.13·77-s + 0.900·79-s + 1.31·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.670369372951772445179851156780, −7.69303769327501289806874681987, −6.81676943247422480081289776053, −6.12168697635411389090145305981, −5.43258380067648598272060672456, −4.84611148766763070327520222578, −3.40955952987947895762360573844, −2.45312259579370360816992630472, −1.86513620948226769101175426335, 0,
1.86513620948226769101175426335, 2.45312259579370360816992630472, 3.40955952987947895762360573844, 4.84611148766763070327520222578, 5.43258380067648598272060672456, 6.12168697635411389090145305981, 6.81676943247422480081289776053, 7.69303769327501289806874681987, 8.670369372951772445179851156780