L(s) = 1 | + 1.41·2-s + (−0.836 − 2.88i)3-s + 2.00·4-s + (−1.18 − 4.07i)6-s + 2.64i·7-s + 2.82·8-s + (−7.60 + 4.81i)9-s + 13.1i·11-s + (−1.67 − 5.76i)12-s − 6.59i·13-s + 3.74i·14-s + 4.00·16-s + 21.7·17-s + (−10.7 + 6.81i)18-s + 5.34·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.278 − 0.960i)3-s + 0.500·4-s + (−0.197 − 0.679i)6-s + 0.377i·7-s + 0.353·8-s + (−0.844 + 0.535i)9-s + 1.19i·11-s + (−0.139 − 0.480i)12-s − 0.507i·13-s + 0.267i·14-s + 0.250·16-s + 1.27·17-s + (−0.597 + 0.378i)18-s + 0.281·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.775276366\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.775276366\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (0.836 + 2.88i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 - 13.1iT - 121T^{2} \) |
| 13 | \( 1 + 6.59iT - 169T^{2} \) |
| 17 | \( 1 - 21.7T + 289T^{2} \) |
| 19 | \( 1 - 5.34T + 361T^{2} \) |
| 23 | \( 1 - 0.0360T + 529T^{2} \) |
| 29 | \( 1 - 0.0748iT - 841T^{2} \) |
| 31 | \( 1 - 36.8T + 961T^{2} \) |
| 37 | \( 1 - 71.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 3.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 93.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 49.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 68.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 1.12iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 70.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 0.397T + 6.88e3T^{2} \) |
| 89 | \( 1 - 1.85iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 34.4iT - 9.40e3T^{2} \) |
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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
−9.923635188028292012964947544275, −8.649711348728730677757193472871, −7.74095341068959572856884679385, −7.16207965155860156737753609196, −6.21717261922811326992840124833, −5.45677372647016556545593900904, −4.65478575336890181380268058668, −3.23343863639041731945870116612, −2.28678126705051002878058746482, −1.10869842136843914732899958624,
0.856522310401162140117166670492, 2.76528158053170355771261137564, 3.65904310068311762307965123276, 4.37272039188628364581542135084, 5.50825190817615953954695472427, 5.94197015915981993280367226286, 7.07334767730419295618593357665, 8.119198355228477103034033021135, 9.014444179097280269444407160091, 9.939449021448767954189836669079