L(s) = 1 | − 8.86·3-s − 7.47i·5-s + 51.5·9-s − 61.2i·11-s − 12.0i·13-s + 66.2i·15-s − 100. i·17-s − 73.6·19-s + 96.8i·23-s + 69.1·25-s − 217.·27-s + 284.·29-s + 256.·31-s + 542. i·33-s + 408.·37-s + ⋯ |
L(s) = 1 | − 1.70·3-s − 0.668i·5-s + 1.90·9-s − 1.67i·11-s − 0.257i·13-s + 1.14i·15-s − 1.43i·17-s − 0.889·19-s + 0.877i·23-s + 0.552·25-s − 1.54·27-s + 1.82·29-s + 1.48·31-s + 2.86i·33-s + 1.81·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9223767409\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9223767409\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 8.86T + 27T^{2} \) |
| 5 | \( 1 + 7.47iT - 125T^{2} \) |
| 11 | \( 1 + 61.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 12.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 100. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 73.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 284.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 256.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 408.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 120. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 308. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 26.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 311.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 444.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 488. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 585. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 416. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 819. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 318. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 218.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 19.4iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 306. iT - 9.12e5T^{2} \) |
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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
−9.741964485636199816175197390828, −8.730852254542766512626819562550, −7.82777330139728795085707492434, −6.54304242919089953763732388166, −6.06104538890923637192878794065, −5.08655614854286561858219609247, −4.53204022806833895916943856628, −2.98861350132794032729388420737, −0.989389585863575014604213873134, −0.45650989955712711569682712677,
1.14434493850927682528585218025, 2.49078069989038618227636816706, 4.43300042513150515813346980228, 4.62091180799437584491442278497, 6.26697503863164572831783098549, 6.36078922247112223610706492772, 7.32335885621288958397159630331, 8.446095077637636634513883466966, 9.924867126912766335201460634955, 10.31887729619846261519879947283