L(s) = 1 | − 9.35·5-s − 21.2i·7-s − 50.9i·11-s + 33.2i·13-s + 120. i·17-s + 82.1·19-s + 95.3·23-s − 37.4·25-s − 83.1·29-s − 36.4i·31-s + 198. i·35-s + 201. i·37-s + 291. i·41-s + 457.·43-s − 628.·47-s + ⋯ |
L(s) = 1 | − 0.837·5-s − 1.14i·7-s − 1.39i·11-s + 0.709i·13-s + 1.72i·17-s + 0.991·19-s + 0.864·23-s − 0.299·25-s − 0.532·29-s − 0.210i·31-s + 0.958i·35-s + 0.896i·37-s + 1.11i·41-s + 1.62·43-s − 1.95·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.616725178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.616725178\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 9.35T + 125T^{2} \) |
| 7 | \( 1 + 21.2iT - 343T^{2} \) |
| 11 | \( 1 + 50.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 33.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 120. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 82.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 95.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 83.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 36.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 201. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 291. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 628.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 659.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 44.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 149. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 134.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 291.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 888.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 714. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 827. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 19.6iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 805.T + 9.12e5T^{2} \) |
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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
−8.686961123027427805604991204462, −8.040596976066384248874039896798, −7.38824068394657168328530857394, −6.51486802557043795884823969486, −5.70606190981758866548937738921, −4.49712986698631528131065744379, −3.79019547154854677111863614660, −3.16687904596614077791396530747, −1.48430273003261156442458109383, −0.55243817258086469057510520913,
0.69550262795321320430317543057, 2.20523520721050160490247364008, 2.98701743487187095477337310004, 4.06743877441634958478514095172, 5.14061851622584611635414064811, 5.51811985236660520837157331335, 6.98134519483033193456218003099, 7.39507495353937672462623268628, 8.185013518398243124754738305834, 9.303231533121618506619916884947