L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−1.64 − 1.50i)5-s + 1.00i·6-s + (2.19 + 2.19i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.0993 + 2.23i)10-s − 2.28i·11-s + (0.707 − 0.707i)12-s + (0.645 + 0.645i)13-s − 3.10i·14-s + (0.0993 + 2.23i)15-s − 1.00·16-s + (1.02 − 1.02i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.737 − 0.675i)5-s + 0.408i·6-s + (0.830 + 0.830i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.0314 + 0.706i)10-s − 0.689i·11-s + (0.204 − 0.204i)12-s + (0.179 + 0.179i)13-s − 0.830i·14-s + (0.0256 + 0.576i)15-s − 0.250·16-s + (0.249 − 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0598 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0598 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.701681 - 0.660899i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.701681 - 0.660899i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.64 + 1.50i)T \) |
| 31 | \( 1 + (-5.39 + 1.36i)T \) |
good | 7 | \( 1 + (-2.19 - 2.19i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.28iT - 11T^{2} \) |
| 13 | \( 1 + (-0.645 - 0.645i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.02 + 1.02i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.57iT - 19T^{2} \) |
| 23 | \( 1 + (-0.646 - 0.646i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 37 | \( 1 + (-5.23 + 5.23i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 + (7.37 + 7.37i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.53 + 2.53i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.61 + 8.61i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.51iT - 59T^{2} \) |
| 61 | \( 1 + 7.53iT - 61T^{2} \) |
| 67 | \( 1 + (0.158 + 0.158i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.65T + 71T^{2} \) |
| 73 | \( 1 + (1.99 + 1.99i)T + 73iT^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + (7.17 + 7.17i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + (4.14 + 4.14i)T + 97iT^{2} \) |
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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
−9.859924442668105986470447322499, −8.825339165230369557692715554217, −8.269935810948941110828876777711, −7.71082326987237242241304354329, −6.46957388478014509928167075515, −5.40168603295810547818571702238, −4.57696262335913361029275592902, −3.33725954639478419052947370384, −1.96477608487039671264912986452, −0.73977331366524703554504116660,
1.07722957823621091749995277487, 2.93392729404849703491852166484, 4.36905565899034291312050218120, 4.77807087116429789310381977488, 6.24487760311087473980491202053, 6.91183230325567817072465441253, 7.81844587247871173934674187696, 8.303803141760941431734293668227, 9.588373270055770163401431963611, 10.31985420574660426406579449994