| L(s) = 1 | − 9.41·5-s + (−3.32 + 6.16i)7-s − 17.6·11-s + 17.9i·13-s − 13.3·17-s − 14.6·19-s + 30.1·23-s + 63.6·25-s − 25.9i·29-s − 11.5·31-s + (31.2 − 57.9i)35-s − 17.9·37-s − 12.6·41-s + 19.2i·43-s + 48.3i·47-s + ⋯ |
| L(s) = 1 | − 1.88·5-s + (−0.474 + 0.880i)7-s − 1.60·11-s + 1.37i·13-s − 0.787·17-s − 0.770·19-s + 1.31·23-s + 2.54·25-s − 0.894i·29-s − 0.372·31-s + (0.893 − 1.65i)35-s − 0.484·37-s − 0.308·41-s + 0.447i·43-s + 1.02i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2870174572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2870174572\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (3.32 - 6.16i)T \) |
| good | 5 | \( 1 + 9.41T + 25T^{2} \) |
| 11 | \( 1 + 17.6T + 121T^{2} \) |
| 13 | \( 1 - 17.9iT - 169T^{2} \) |
| 17 | \( 1 + 13.3T + 289T^{2} \) |
| 19 | \( 1 + 14.6T + 361T^{2} \) |
| 23 | \( 1 - 30.1T + 529T^{2} \) |
| 29 | \( 1 + 25.9iT - 841T^{2} \) |
| 31 | \( 1 + 11.5T + 961T^{2} \) |
| 37 | \( 1 + 17.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 12.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 19.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 48.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 3.42iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 53.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 59.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 104. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 76.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 142. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 47.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 90.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 122.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 94.7iT - 9.40e3T^{2} \) |
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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
−9.467883163376812888698271342159, −8.665815814995164159967240473778, −8.071728425264043641616764211110, −7.18020306781985591084163678940, −6.43666385352776524023621224758, −5.06023466004337580584426953717, −4.37402838679028011340709991964, −3.29724881292894814711839786563, −2.32284055648905404292043010854, −0.16472072673298664473095549022,
0.59756912784550284259750988393, 2.85252565202800826819304193522, 3.54969529894960556763898531542, 4.54777087925235851832844259631, 5.36334216129204565579533937070, 6.93075967289556370046974231935, 7.35546759934299022995230878608, 8.178608436896741309695788400140, 8.736626457853485337276063160691, 10.24709415315303505291742112652
