| L(s) = 1 | − 12.8·5-s + 36.8·11-s − 87.1·13-s + 102.·17-s − 95.8·19-s + 96·23-s + 39.4·25-s + 212.·29-s + 159.·31-s + 128.·37-s − 298.·41-s − 33.3·43-s + 271.·47-s − 448.·53-s − 472.·55-s − 668.·59-s + 243.·61-s + 1.11e3·65-s − 335.·67-s + 339.·71-s + 918.·73-s − 136.·79-s + 287.·83-s − 1.31e3·85-s + 161.·89-s + 1.22e3·95-s − 182.·97-s + ⋯ |
| L(s) = 1 | − 1.14·5-s + 1.00·11-s − 1.85·13-s + 1.46·17-s − 1.15·19-s + 0.870·23-s + 0.315·25-s + 1.35·29-s + 0.922·31-s + 0.571·37-s − 1.13·41-s − 0.118·43-s + 0.841·47-s − 1.16·53-s − 1.15·55-s − 1.47·59-s + 0.511·61-s + 2.13·65-s − 0.611·67-s + 0.567·71-s + 1.47·73-s − 0.194·79-s + 0.380·83-s − 1.67·85-s + 0.192·89-s + 1.32·95-s − 0.191·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 12.8T + 125T^{2} \) |
| 11 | \( 1 - 36.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96T + 1.21e4T^{2} \) |
| 29 | \( 1 - 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 448.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 668.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 335.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 918.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 136.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 161.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 182.T + 9.12e5T^{2} \) |
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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
−8.370677412707212591804217431868, −7.79494081819462023944810151523, −7.03343952562559409027075174097, −6.31227802091602216399225308992, −5.01678580370133048506206730101, −4.43740789026320731040443953417, −3.48422414253991090124521783926, −2.57154946108970791892650913271, −1.12304985242660808172740169271, 0,
1.12304985242660808172740169271, 2.57154946108970791892650913271, 3.48422414253991090124521783926, 4.43740789026320731040443953417, 5.01678580370133048506206730101, 6.31227802091602216399225308992, 7.03343952562559409027075174097, 7.79494081819462023944810151523, 8.370677412707212591804217431868
