L(s) = 1 | − 10i·2-s + 14i·3-s − 68·4-s + 140·6-s − 49i·7-s + 360i·8-s + 47·9-s + 232·11-s − 952i·12-s + 140i·13-s − 490·14-s + 1.42e3·16-s − 1.72e3i·17-s − 470i·18-s + 98·19-s + ⋯ |
L(s) = 1 | − 1.76i·2-s + 0.898i·3-s − 2.12·4-s + 1.58·6-s − 0.377i·7-s + 1.98i·8-s + 0.193·9-s + 0.578·11-s − 1.90i·12-s + 0.229i·13-s − 0.668·14-s + 1.39·16-s − 1.44i·17-s − 0.341i·18-s + 0.0622·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9810111858\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9810111858\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + 49iT \) |
good | 2 | \( 1 + 10iT - 32T^{2} \) |
| 3 | \( 1 - 14iT - 243T^{2} \) |
| 11 | \( 1 - 232T + 1.61e5T^{2} \) |
| 13 | \( 1 - 140iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.72e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 98T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.82e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.03e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.08e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.32e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.26e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.73e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.84e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.80e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.05e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.85e4iT - 8.58e9T^{2} \) |
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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
−11.12417186135891365996708532963, −10.40381455646279359682469957744, −9.497692167891142192140958305570, −8.960262953091358381471642166789, −7.16836310586309377145304452422, −5.13264298361291426951158643052, −4.17507049099793042123017350182, −3.32512056566408719348114438354, −1.81701390460019766170532608122, −0.32602610511261903622463805952,
1.52607186214411363042823684444, 3.90815879922517592590090250701, 5.38038814576310393891059234361, 6.31026300298808922525380989898, 7.10463520403413235450155151994, 8.013606675557466591342558635490, 8.829539497390668569175572792110, 10.00352208368061456795185530937, 11.68674291682769855490633286579, 12.91269464340591816008602117613