L(s) = 1 | − 5-s − 11-s + 6·13-s + 2·17-s + 4·19-s − 8·23-s + 25-s − 2·29-s + 4·31-s − 6·37-s − 6·41-s + 4·43-s + 8·47-s − 2·53-s + 55-s + 12·59-s + 10·61-s − 6·65-s − 16·67-s − 10·73-s + 12·79-s + 16·83-s − 2·85-s + 10·89-s − 4·95-s + 2·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.274·53-s + 0.134·55-s + 1.56·59-s + 1.28·61-s − 0.744·65-s − 1.95·67-s − 1.17·73-s + 1.35·79-s + 1.75·83-s − 0.216·85-s + 1.05·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.846493429\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.846493429\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.16853330793117, −11.97917767480289, −11.78951437658527, −10.99476348674619, −10.73239504933272, −10.24847712342516, −9.820375259064591, −9.277677578462074, −8.707930121574158, −8.341006071828294, −8.006411085996875, −7.390056838826979, −7.106046644623890, −6.189554799488913, −6.179131125060797, −5.418504497991610, −5.146000279817503, −4.254982329057765, −3.978782219243055, −3.399024113359337, −3.093422432713664, −2.206236340346657, −1.730166821587507, −0.9877707663702030, −0.5042765365483763,
0.5042765365483763, 0.9877707663702030, 1.730166821587507, 2.206236340346657, 3.093422432713664, 3.399024113359337, 3.978782219243055, 4.254982329057765, 5.146000279817503, 5.418504497991610, 6.179131125060797, 6.189554799488913, 7.106046644623890, 7.390056838826979, 8.006411085996875, 8.341006071828294, 8.707930121574158, 9.277677578462074, 9.820375259064591, 10.24847712342516, 10.73239504933272, 10.99476348674619, 11.78951437658527, 11.97917767480289, 12.16853330793117