L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (0.975 − 2.01i)5-s − 1.00i·6-s + (−3.17 − 3.17i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (2.11 − 0.732i)10-s + 5.75i·11-s + (0.707 − 0.707i)12-s + (−0.935 − 0.935i)13-s − 4.48i·14-s + (−2.11 + 0.732i)15-s − 1.00·16-s + (2.02 − 2.02i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.436 − 0.899i)5-s − 0.408i·6-s + (−1.19 − 1.19i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.668 − 0.231i)10-s + 1.73i·11-s + (0.204 − 0.204i)12-s + (−0.259 − 0.259i)13-s − 1.19i·14-s + (−0.545 + 0.189i)15-s − 0.250·16-s + (0.490 − 0.490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.401420 - 0.777403i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.401420 - 0.777403i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.975 + 2.01i)T \) |
| 31 | \( 1 + (3.66 - 4.18i)T \) |
good | 7 | \( 1 + (3.17 + 3.17i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.75iT - 11T^{2} \) |
| 13 | \( 1 + (0.935 + 0.935i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.02 + 2.02i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.40iT - 19T^{2} \) |
| 23 | \( 1 + (4.01 + 4.01i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 37 | \( 1 + (3.68 - 3.68i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.446T + 41T^{2} \) |
| 43 | \( 1 + (8.48 + 8.48i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.10 + 7.10i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.35 - 2.35i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.60iT - 59T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 + (-6.43 - 6.43i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.97T + 71T^{2} \) |
| 73 | \( 1 + (-3.59 - 3.59i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 + (2.02 + 2.02i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + (-13.2 - 13.2i)T + 97iT^{2} \) |
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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
−9.874251350865824348365967357179, −8.984765226762126425557329628701, −7.76118114168644732767055365455, −6.95135584752027698584026089164, −6.60262062984171882759675590683, −5.24298612257069370283159763276, −4.70797366779338112028296349942, −3.60756643579562609335549751931, −2.07457444221175936058633899172, −0.34089568614918316617201353382,
2.00679099887305841685741935016, 3.37342829421532010810732172813, 3.53608013405634410086725460879, 5.46832938699366459799044400697, 5.96809560936225178144198853157, 6.34671458302510498640723337637, 7.86106786829021940704153138302, 9.055388668150647230038688560816, 9.752431328550335599582919974352, 10.32001225193895441062773951448