L(s) = 1 | + 2·7-s − 6·11-s − 2·13-s − 3·17-s + 19-s + 9·23-s − 3·29-s + 4·31-s + 7·37-s − 6·41-s − 10·43-s − 9·47-s − 3·49-s − 12·53-s − 15·59-s + 8·61-s + 67-s − 12·71-s − 11·73-s − 12·77-s + 4·79-s + 12·83-s − 15·89-s − 4·91-s − 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 1.80·11-s − 0.554·13-s − 0.727·17-s + 0.229·19-s + 1.87·23-s − 0.557·29-s + 0.718·31-s + 1.15·37-s − 0.937·41-s − 1.52·43-s − 1.31·47-s − 3/7·49-s − 1.64·53-s − 1.95·59-s + 1.02·61-s + 0.122·67-s − 1.42·71-s − 1.28·73-s − 1.36·77-s + 0.450·79-s + 1.31·83-s − 1.58·89-s − 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4960988213\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4960988213\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.02444759692129, −12.48811358853895, −11.92220605371381, −11.36473808492040, −11.03489525064734, −10.70644119036229, −10.12077403474570, −9.622437275832480, −9.249203322402944, −8.493550170996438, −8.139953014214663, −7.832841494994322, −7.231615660126282, −6.797799118834812, −6.242217836741681, −5.503474944321690, −5.081406411562330, −4.727096002262346, −4.428420293852125, −3.291283782994676, −3.006215538242159, −2.496589434970089, −1.754682507983146, −1.266667046673950, −0.1881791040386955,
0.1881791040386955, 1.266667046673950, 1.754682507983146, 2.496589434970089, 3.006215538242159, 3.291283782994676, 4.428420293852125, 4.727096002262346, 5.081406411562330, 5.503474944321690, 6.242217836741681, 6.797799118834812, 7.231615660126282, 7.832841494994322, 8.139953014214663, 8.493550170996438, 9.249203322402944, 9.622437275832480, 10.12077403474570, 10.70644119036229, 11.03489525064734, 11.36473808492040, 11.92220605371381, 12.48811358853895, 13.02444759692129