| L(s) = 1 | + 12·3-s + 54·5-s + 88·7-s − 99·9-s − 540·11-s − 418·13-s + 648·15-s + 594·17-s − 836·19-s + 1.05e3·21-s + 4.10e3·23-s − 209·25-s − 4.10e3·27-s − 594·29-s − 4.25e3·31-s − 6.48e3·33-s + 4.75e3·35-s − 298·37-s − 5.01e3·39-s + 1.72e4·41-s + 1.21e4·43-s − 5.34e3·45-s + 1.29e3·47-s − 9.06e3·49-s + 7.12e3·51-s + 1.94e4·53-s − 2.91e4·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.965·5-s + 0.678·7-s − 0.407·9-s − 1.34·11-s − 0.685·13-s + 0.743·15-s + 0.498·17-s − 0.531·19-s + 0.522·21-s + 1.61·23-s − 0.0668·25-s − 1.08·27-s − 0.131·29-s − 0.795·31-s − 1.03·33-s + 0.655·35-s − 0.0357·37-s − 0.528·39-s + 1.60·41-s + 0.997·43-s − 0.393·45-s + 0.0855·47-s − 0.539·49-s + 0.383·51-s + 0.953·53-s − 1.29·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.729941294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.729941294\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 7 | \( 1 - 88 T + p^{5} T^{2} \) |
| 11 | \( 1 + 540 T + p^{5} T^{2} \) |
| 13 | \( 1 + 418 T + p^{5} T^{2} \) |
| 17 | \( 1 - 594 T + p^{5} T^{2} \) |
| 19 | \( 1 + 44 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 4104 T + p^{5} T^{2} \) |
| 29 | \( 1 + 594 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4256 T + p^{5} T^{2} \) |
| 37 | \( 1 + 298 T + p^{5} T^{2} \) |
| 41 | \( 1 - 17226 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12100 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1296 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19494 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7668 T + p^{5} T^{2} \) |
| 61 | \( 1 + 34738 T + p^{5} T^{2} \) |
| 67 | \( 1 + 21812 T + p^{5} T^{2} \) |
| 71 | \( 1 - 46872 T + p^{5} T^{2} \) |
| 73 | \( 1 - 67562 T + p^{5} T^{2} \) |
| 79 | \( 1 - 76912 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 29754 T + p^{5} T^{2} \) |
| 97 | \( 1 + 122398 T + p^{5} T^{2} \) |
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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
−18.04383618758266999358334523049, −16.94998892060282683766285434716, −15.08703638296098589248965903262, −14.09667517674615670738313853545, −12.89539891565161236343903424380, −10.83048349303421078800041691529, −9.281178657875979264696140206247, −7.77817942545382928329939365464, −5.38115118881380520172416574908, −2.47443382434278018679916361115,
2.47443382434278018679916361115, 5.38115118881380520172416574908, 7.77817942545382928329939365464, 9.281178657875979264696140206247, 10.83048349303421078800041691529, 12.89539891565161236343903424380, 14.09667517674615670738313853545, 15.08703638296098589248965903262, 16.94998892060282683766285434716, 18.04383618758266999358334523049
