L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1.5 − 2.59i)5-s + 0.999·6-s − 0.999·8-s + (1 + 1.73i)9-s + (1.5 − 2.59i)10-s + (−3 + 5.19i)11-s + (0.499 + 0.866i)12-s − 13-s − 3·15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (−1 + 1.73i)18-s + (1 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s + 0.408·6-s − 0.353·8-s + (0.333 + 0.577i)9-s + (0.474 − 0.821i)10-s + (−0.904 + 1.56i)11-s + (0.144 + 0.249i)12-s − 0.277·13-s − 0.774·15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (−0.235 + 0.408i)18-s + (0.229 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.315068226\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.315068226\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.845740764750260943765110075727, −8.705247151586912474214417468958, −8.158892123918168815948896961234, −7.49968016997481774375218022745, −6.89556707169694260702814700494, −5.58622667962240967415124992645, −4.61769563116742785332745278460, −4.39720952056076697065234897029, −2.74466515892824068550522721942, −1.49758024445134815192422686207,
0.48091263426210841778294514723, 2.62159637039693162580854411518, 3.17898342347373831089946990528, 3.94380659734559352484438786991, 4.98731470874933024597318208678, 6.06814016463068577603369666091, 6.93171324093734888688570148228, 7.84774010876690801876079599227, 8.769635522376092508565784619068, 9.606587567387092992519671818434