L(s) = 1 | + 3.47·5-s − i·7-s + 5.11i·11-s − 0.268i·13-s − 1.96i·17-s − 0.909·19-s − 5.86·23-s + 7.04·25-s − 7.19·29-s + 4.57i·31-s − 3.47i·35-s + 6.98i·37-s + 8.34i·41-s + 9.25·43-s + 6.49·47-s + ⋯ |
L(s) = 1 | + 1.55·5-s − 0.377i·7-s + 1.54i·11-s − 0.0744i·13-s − 0.476i·17-s − 0.208·19-s − 1.22·23-s + 1.40·25-s − 1.33·29-s + 0.821i·31-s − 0.586i·35-s + 1.14i·37-s + 1.30i·41-s + 1.41·43-s + 0.947·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.336923913\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.336923913\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
good | 5 | \( 1 - 3.47T + 5T^{2} \) |
| 11 | \( 1 - 5.11iT - 11T^{2} \) |
| 13 | \( 1 + 0.268iT - 13T^{2} \) |
| 17 | \( 1 + 1.96iT - 17T^{2} \) |
| 19 | \( 1 + 0.909T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + 7.19T + 29T^{2} \) |
| 31 | \( 1 - 4.57iT - 31T^{2} \) |
| 37 | \( 1 - 6.98iT - 37T^{2} \) |
| 41 | \( 1 - 8.34iT - 41T^{2} \) |
| 43 | \( 1 - 9.25T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 - 4.54T + 53T^{2} \) |
| 59 | \( 1 - 3.50iT - 59T^{2} \) |
| 61 | \( 1 - 1.96iT - 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 + 9.01T + 73T^{2} \) |
| 79 | \( 1 - 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 4.96iT - 83T^{2} \) |
| 89 | \( 1 - 18.2iT - 89T^{2} \) |
| 97 | \( 1 - 3.39T + 97T^{2} \) |
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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
−8.167916326802079465281468054700, −7.36629900974614057918586949678, −6.80552684618022429831428311193, −6.05818850538427299330315398198, −5.40669745777288526758644085900, −4.67209641071654296983720762384, −3.92270871603351107504592699145, −2.66439047583172196799141234196, −2.04759227458208773932243794723, −1.24474889543280377245687268840,
0.56020638836161763193599954516, 1.90178841952352067613315707185, 2.34081810493461459394888487292, 3.45150567927644784982454399148, 4.21023217846903590130260966782, 5.52779452876710739416908412711, 5.79170619811245771021539635940, 6.14362849296650178206881372942, 7.19485072726364839570167581078, 8.012215192472246057696012676755