L(s) = 1 | − 3·7-s − 4·11-s − 7·13-s − 4·17-s − 19-s + 8·23-s − 3·31-s + 2·37-s + 6·41-s − 11·43-s − 6·47-s + 2·49-s − 6·53-s + 6·59-s − 61-s − 15·67-s − 6·71-s + 2·73-s + 12·77-s − 8·79-s − 2·83-s + 16·89-s + 21·91-s + 13·97-s + 2·101-s − 16·103-s + 18·107-s + ⋯ |
L(s) = 1 | − 1.13·7-s − 1.20·11-s − 1.94·13-s − 0.970·17-s − 0.229·19-s + 1.66·23-s − 0.538·31-s + 0.328·37-s + 0.937·41-s − 1.67·43-s − 0.875·47-s + 2/7·49-s − 0.824·53-s + 0.781·59-s − 0.128·61-s − 1.83·67-s − 0.712·71-s + 0.234·73-s + 1.36·77-s − 0.900·79-s − 0.219·83-s + 1.69·89-s + 2.20·91-s + 1.31·97-s + 0.199·101-s − 1.57·103-s + 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5511855485\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5511855485\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.73518458984781631479938463846, −7.21525048272757119527735815052, −6.65895029197279271200590164901, −5.84004587447743808171476362032, −4.91105587842042433939911556532, −4.64014047694129360989620976042, −3.28177997650748355599179591105, −2.80452156426397466716492169462, −2.01960379496755926227233066072, −0.34627339890746046357547877726,
0.34627339890746046357547877726, 2.01960379496755926227233066072, 2.80452156426397466716492169462, 3.28177997650748355599179591105, 4.64014047694129360989620976042, 4.91105587842042433939911556532, 5.84004587447743808171476362032, 6.65895029197279271200590164901, 7.21525048272757119527735815052, 7.73518458984781631479938463846