L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−1.89 − 1.18i)5-s + (0.705 − 0.705i)7-s + (0.707 − 0.707i)8-s + (0.498 + 2.17i)10-s − 1.32·11-s + (−0.741 + 0.741i)13-s − 0.997·14-s − 1.00·16-s + (−2.17 + 2.17i)17-s + (0.939 + 4.25i)19-s + (1.18 − 1.89i)20-s + (0.939 + 0.939i)22-s + (1.08 + 1.08i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.846 − 0.531i)5-s + (0.266 − 0.266i)7-s + (0.250 − 0.250i)8-s + (0.157 + 0.689i)10-s − 0.400·11-s + (−0.205 + 0.205i)13-s − 0.266·14-s − 0.250·16-s + (−0.527 + 0.527i)17-s + (0.215 + 0.976i)19-s + (0.265 − 0.423i)20-s + (0.200 + 0.200i)22-s + (0.225 + 0.225i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9289983629\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9289983629\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1.89 + 1.18i)T \) |
| 19 | \( 1 + (-0.939 - 4.25i)T \) |
good | 7 | \( 1 + (-0.705 + 0.705i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + (0.741 - 0.741i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.17 - 2.17i)T - 17iT^{2} \) |
| 23 | \( 1 + (-1.08 - 1.08i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 + 5.95iT - 31T^{2} \) |
| 37 | \( 1 + (1.33 + 1.33i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.531iT - 41T^{2} \) |
| 43 | \( 1 + (-1.53 - 1.53i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.13 - 4.13i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.48 + 5.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 + (1.99 + 1.99i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.71iT - 71T^{2} \) |
| 73 | \( 1 + (5.83 + 5.83i)T + 73iT^{2} \) |
| 79 | \( 1 - 9.31T + 79T^{2} \) |
| 83 | \( 1 + (-9.50 - 9.50i)T + 83iT^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + (-1.11 - 1.11i)T + 97iT^{2} \) |
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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
−9.292112133628018188281947288162, −8.470343083696243784342939122503, −7.916563786422326969195633952612, −7.26546223269409868359412251606, −6.16128571897205366671387497469, −4.97903852703752398203285099902, −4.20875307176276853061250913825, −3.38800656323186411366555939328, −2.11251841491107657822581339742, −0.869123571132206243527211192525,
0.58360206796172013868757043488, 2.34465984433537219578781761998, 3.27543169793386069089361142916, 4.61538469197169753933657912451, 5.18157834458560141633104515876, 6.44972005345423841431896117798, 7.03313606226755497845504637323, 7.73309661999335786145720567859, 8.546191874216561789660043372969, 9.071936555028714416006390060763