L(s) = 1 | − 0.717·5-s − 3i·11-s − 2.44i·13-s − 5.91·17-s − 5.91i·19-s − 4.24i·23-s − 4.48·25-s + 7.24i·29-s + 9.08i·31-s − 0.242·37-s − 11.8·41-s + 0.242·43-s + 5.91·47-s + 7.24i·53-s + 2.15i·55-s + ⋯ |
L(s) = 1 | − 0.320·5-s − 0.904i·11-s − 0.679i·13-s − 1.43·17-s − 1.35i·19-s − 0.884i·23-s − 0.897·25-s + 1.34i·29-s + 1.63i·31-s − 0.0398·37-s − 1.84·41-s + 0.0370·43-s + 0.862·47-s + 0.994i·53-s + 0.290i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5788890273\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5788890273\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 0.717T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 5.91iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 - 7.24iT - 29T^{2} \) |
| 31 | \( 1 - 9.08iT - 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.242T + 43T^{2} \) |
| 47 | \( 1 - 5.91T + 47T^{2} \) |
| 53 | \( 1 - 7.24iT - 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 + 1.01iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 1.75iT - 71T^{2} \) |
| 73 | \( 1 + 1.43iT - 73T^{2} \) |
| 79 | \( 1 - 2.75T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−8.281589201091906526897274268462, −7.34680536411697831375336145697, −6.73892045751274329347174134811, −6.15760656576693218003371549796, −5.12023758372681768627474554002, −4.72862739998885051506780539449, −3.65911018945373437650200803292, −3.03357264331842822635424749992, −2.14603077676094581677458177926, −0.870513774538379316294797767107,
0.16502663567503521367594274302, 1.82001496355857464923156134461, 2.19171934430473069091317529684, 3.57934170893770712271425596789, 4.12930401395614195573500873948, 4.75071989638265962076183092630, 5.75381179956928472447128937431, 6.35777073938391810427100196228, 7.11768070555094674380552672389, 7.76453589063073225818866371570