| L(s) = 1 | − 1.02e3·4-s + (1.97e3 + 2.42e3i)5-s + 8.48e3i·7-s − 5.90e4·9-s − 2.03e5·11-s + 1.04e6·16-s − 1.85e6i·17-s − 2.47e6·19-s + (−2.02e6 − 2.47e6i)20-s + 1.18e7i·23-s + (−1.96e6 + 9.56e6i)25-s − 8.69e6i·28-s + (−2.05e7 + 1.67e7i)35-s + 6.04e7·36-s − 2.03e8i·43-s + 2.08e8·44-s + ⋯ |
| L(s) = 1 | − 4-s + (0.632 + 0.774i)5-s + 0.504i·7-s − 0.999·9-s − 1.26·11-s + 16-s − 1.30i·17-s − 19-s + (−0.632 − 0.774i)20-s + 1.83i·23-s + (−0.200 + 0.979i)25-s − 0.504i·28-s + (−0.391 + 0.319i)35-s + 0.999·36-s − 1.38i·43-s + 1.26·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.612044 - 0.290556i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612044 - 0.290556i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.97e3 - 2.42e3i)T \) |
| 19 | \( 1 + 2.47e6T \) |
| good | 2 | \( 1 + 1.02e3T^{2} \) |
| 3 | \( 1 + 5.90e4T^{2} \) |
| 7 | \( 1 - 8.48e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.03e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.85e6iT - 2.01e12T^{2} \) |
| 23 | \( 1 - 1.18e7iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 4.20e14T^{2} \) |
| 31 | \( 1 - 8.19e14T^{2} \) |
| 37 | \( 1 + 4.80e15T^{2} \) |
| 41 | \( 1 - 1.34e16T^{2} \) |
| 43 | \( 1 + 2.03e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 4.28e7iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 1.74e17T^{2} \) |
| 59 | \( 1 - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.60e9T + 7.13e17T^{2} \) |
| 67 | \( 1 + 1.82e18T^{2} \) |
| 71 | \( 1 - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.70e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 9.46e18T^{2} \) |
| 83 | \( 1 + 7.57e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 3.11e19T^{2} \) |
| 97 | \( 1 + 7.37e19T^{2} \) |
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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
−11.77719259641969960674549637429, −10.63396366213544073879169660943, −9.600166445566417064624825380321, −8.676299838494493369399258466472, −7.45176617241802238810016800197, −5.80047099553414431823568676264, −5.14378654070056368628474526800, −3.31919316396098611609328388716, −2.27361095777284880183677926131, −0.24820434471412867367521316336,
0.75020506806192037575935143114, 2.42135042513146881104795493235, 4.13208760491157723585707550606, 5.15730103531334606385314228817, 6.17217720397381422258065391455, 8.231373408226320965486263192293, 8.595756068064213234994245972359, 9.991202521767680121983389418604, 10.76794745697115588505466805548, 12.60068889380601118170171174584
