| L(s) = 1 | + 15.7·3-s − 70.8i·5-s + (78.9 + 102. i)7-s + 5.62·9-s + 279. i·11-s + 763. i·13-s − 1.11e3i·15-s + 1.07e3i·17-s − 2.18e3·19-s + (1.24e3 + 1.62e3i)21-s − 2.26e3i·23-s − 1.89e3·25-s − 3.74e3·27-s − 463.·29-s − 6.23e3·31-s + ⋯ |
| L(s) = 1 | + 1.01·3-s − 1.26i·5-s + (0.609 + 0.793i)7-s + 0.0231·9-s + 0.695i·11-s + 1.25i·13-s − 1.28i·15-s + 0.902i·17-s − 1.39·19-s + (0.616 + 0.802i)21-s − 0.892i·23-s − 0.605·25-s − 0.988·27-s − 0.102·29-s − 1.16·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.793i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.253618437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253618437\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-78.9 - 102. i)T \) |
| good | 3 | \( 1 - 15.7T + 243T^{2} \) |
| 5 | \( 1 + 70.8iT - 3.12e3T^{2} \) |
| 11 | \( 1 - 279. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 763. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.07e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.26e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 463.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.83e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 6.78e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 9.96e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.09e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.34e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.90e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.88e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.90e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 1.66e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.89e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 4.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.71e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 7.41e4iT - 8.58e9T^{2} \) |
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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
−10.50963652307037490868368042860, −9.260839701319141767823872830298, −8.650275738852112869545969305035, −8.428120380467219178308380503087, −7.08692431498629259979913088112, −5.77835433795942804062369231416, −4.70562273020161475395086648371, −3.91899216453805613293675456945, −2.21092670473250159613952859910, −1.69806341699913998415527866011,
0.22450220753325397258577134573, 1.93402874123433869940155567008, 3.10103826237411016959402829067, 3.60385085675287184323728106515, 5.16404454960840556195743170280, 6.40749419514114773355703919686, 7.44975028295280952753532566396, 8.013857397228557979441172792850, 8.957999105476232232296187367487, 10.09224668426076938143553057197
