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
−10.32001225193895441062773951448, −9.752431328550335599582919974352, −9.055388668150647230038688560816, −7.86106786829021940704153138302, −6.34671458302510498640723337637, −5.96809560936225178144198853157, −5.46832938699366459799044400697, −3.53608013405634410086725460879, −3.37342829421532010810732172813, −2.00679099887305841685741935016,
0.34089568614918316617201353382, 2.07457444221175936058633899172, 3.60756643579562609335549751931, 4.70797366779338112028296349942, 5.24298612257069370283159763276, 6.60262062984171882759675590683, 6.95135584752027698584026089164, 7.76118114168644732767055365455, 8.984765226762126425557329628701, 9.874251350865824348365967357179