L(s) = 1 | + (2.12 − 1.22i)2-s + (−1.35 + 0.780i)3-s + (2.01 − 3.49i)4-s + (−0.746 − 2.10i)5-s + (−1.91 + 3.32i)6-s + 4.50i·7-s − 4.99i·8-s + (−0.281 + 0.487i)9-s + (−4.17 − 3.56i)10-s + 2.19·11-s + 6.29i·12-s + (−3.25 − 1.87i)13-s + (5.53 + 9.58i)14-s + (2.65 + 2.26i)15-s + (−2.09 − 3.63i)16-s + (0.576 − 0.332i)17-s + ⋯ |
L(s) = 1 | + (1.50 − 0.868i)2-s + (−0.780 + 0.450i)3-s + (1.00 − 1.74i)4-s + (−0.333 − 0.942i)5-s + (−0.782 + 1.35i)6-s + 1.70i·7-s − 1.76i·8-s + (−0.0938 + 0.162i)9-s + (−1.32 − 1.12i)10-s + 0.662·11-s + 1.81i·12-s + (−0.902 − 0.521i)13-s + (1.47 + 2.56i)14-s + (0.685 + 0.585i)15-s + (−0.524 − 0.909i)16-s + (0.139 − 0.0807i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41323 - 0.690390i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41323 - 0.690390i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.746 + 2.10i)T \) |
| 19 | \( 1 + (3.79 + 2.13i)T \) |
good | 2 | \( 1 + (-2.12 + 1.22i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.35 - 0.780i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 4.50iT - 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 + (3.25 + 1.87i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.576 + 0.332i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.422 + 0.244i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.79 + 3.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 + 3.01iT - 37T^{2} \) |
| 41 | \( 1 + (0.0362 + 0.0627i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.364 - 0.210i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.34 + 2.51i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.26 - 1.30i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.26 - 10.8i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.53 - 6.11i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.95 + 2.86i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.48 + 6.03i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.56 + 1.47i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 + (0.668 - 1.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.79 + 2.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−13.54904110721156554360182732981, −12.34206351070908825367670098949, −12.07074417532374747816088500517, −11.22131451057878586547291674209, −9.878633259856318974038178111203, −8.513808012289403134640464914099, −6.06829029141018733049171405484, −5.24203386977691064781244507407, −4.43271681174362388656045757174, −2.49291675999361816522359616912,
3.53870609946624462287617565049, 4.60996621439293570595761008059, 6.39519176908224723084108393675, 6.78628059402134025531291995615, 7.73875488067375386006310771410, 10.23600703391980731584672672413, 11.43458509967466918818627355672, 12.18695671626136695852509007225, 13.30502861113855239301500948689, 14.37800296510128275719967822463