L(s) = 1 | + (−1.98 + 4.80i)3-s + 11.9·5-s + 22.6i·7-s + (−19.1 − 19.0i)9-s + 61.9i·11-s − 71.3i·13-s + (−23.7 + 57.5i)15-s + 74.3i·17-s − 108.·19-s + (−109. − 44.9i)21-s + 24.7·23-s + 18.6·25-s + (129. − 54.2i)27-s − 84.1·29-s − 130. i·31-s + ⋯ |
L(s) = 1 | + (−0.381 + 0.924i)3-s + 1.07·5-s + 1.22i·7-s + (−0.709 − 0.705i)9-s + 1.69i·11-s − 1.52i·13-s + (−0.408 + 0.990i)15-s + 1.06i·17-s − 1.30·19-s + (−1.13 − 0.467i)21-s + 0.223·23-s + 0.149·25-s + (0.922 − 0.386i)27-s − 0.538·29-s − 0.755i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.036979248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.036979248\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.98 - 4.80i)T \) |
good | 5 | \( 1 - 11.9T + 125T^{2} \) |
| 7 | \( 1 - 22.6iT - 343T^{2} \) |
| 11 | \( 1 - 61.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 71.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 74.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 84.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 130. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 397. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 104. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 161.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 136.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 243. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 358. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 449.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 925.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 55.9iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 928. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 853. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 714.T + 9.12e5T^{2} \) |
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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
−10.13505269710723256521806325938, −9.826210933626906129595855043183, −8.889669482163058996166205796589, −8.091342219830606735701888998372, −6.58142423221495891025189912736, −5.79690433996799643040488373530, −5.22461970578217859458393305025, −4.19673289869938970513526147640, −2.77491228300247058232908386561, −1.84480208834634286571844244490,
0.27988444610127479548540420937, 1.38165751461808270670782353468, 2.44418652586860445113843488097, 3.87253257405995915929781466246, 5.14176475302636496852375722403, 6.13220308161569611661002414085, 6.69706521873764263029694560458, 7.51102420554399040339750192183, 8.653579339181955144235491714714, 9.333736149276368120219841375632