L(s) = 1 | + (−0.551 − 1.30i)2-s + (−1.16 − 1.16i)3-s + (−1.39 + 1.43i)4-s + (0.707 − 0.707i)5-s + (−0.873 + 2.15i)6-s − i·7-s + (2.63 + 1.02i)8-s − 0.295i·9-s + (−1.31 − 0.531i)10-s + (3.97 − 3.97i)11-s + (3.28 − 0.0508i)12-s + (−1.48 − 1.48i)13-s + (−1.30 + 0.551i)14-s − 1.64·15-s + (−0.123 − 3.99i)16-s − 6.33·17-s + ⋯ |
L(s) = 1 | + (−0.389 − 0.920i)2-s + (−0.671 − 0.671i)3-s + (−0.696 + 0.717i)4-s + (0.316 − 0.316i)5-s + (−0.356 + 0.879i)6-s − 0.377i·7-s + (0.932 + 0.361i)8-s − 0.0986i·9-s + (−0.414 − 0.167i)10-s + (1.19 − 1.19i)11-s + (0.949 − 0.0146i)12-s + (−0.413 − 0.413i)13-s + (−0.348 + 0.147i)14-s − 0.424·15-s + (−0.0309 − 0.999i)16-s − 1.53·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.130670 + 0.714466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.130670 + 0.714466i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.551 + 1.30i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (1.16 + 1.16i)T + 3iT^{2} \) |
| 11 | \( 1 + (-3.97 + 3.97i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.48 + 1.48i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.33T + 17T^{2} \) |
| 19 | \( 1 + (-4.80 - 4.80i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.65iT - 23T^{2} \) |
| 29 | \( 1 + (4.24 + 4.24i)T + 29iT^{2} \) |
| 31 | \( 1 + 0.0294T + 31T^{2} \) |
| 37 | \( 1 + (5.73 - 5.73i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.919iT - 41T^{2} \) |
| 43 | \( 1 + (3.37 - 3.37i)T - 43iT^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + (-0.0243 + 0.0243i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.91 - 5.91i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.34 - 1.34i)T + 61iT^{2} \) |
| 67 | \( 1 + (9.68 + 9.68i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.0iT - 71T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 4.67T + 79T^{2} \) |
| 83 | \( 1 + (-4.38 - 4.38i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 0.578T + 97T^{2} \) |
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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
−10.45802633761800481483510167111, −9.388534014414842518146457786488, −8.756947065397113874130509746127, −7.68770267280053047866746894273, −6.60924500293510536199883731913, −5.72791573523272358609166154248, −4.37737337560526486238369298828, −3.29379379527411350855313948600, −1.66213167720764175033259504099, −0.53549327087466945448573970470,
1.93618632378032081173149910630, 4.10101002045737889682284251348, 4.93793373884701930039831101334, 5.72124351918688857947844621781, 6.93091886510795508534369642788, 7.26682006295779150047383160063, 8.999138936915862356801377082179, 9.305754601697503431810019497795, 10.17680276508384595135447746993, 11.11073112969845674479370776130