L(s) = 1 | + (−2.08 − 2.08i)3-s + (−2.48 − 2.48i)5-s + (−0.409 + 0.409i)7-s + 5.68i·9-s + (−3.51 + 3.51i)11-s + 4.68·13-s + 10.3i·15-s + (−2.19 − 3.48i)17-s + 7.51i·19-s + 1.70·21-s + (1.83 − 1.83i)23-s + 7.39i·25-s + (5.59 − 5.59i)27-s + (0.196 + 0.196i)29-s + (−5.18 − 5.18i)31-s + ⋯ |
L(s) = 1 | + (−1.20 − 1.20i)3-s + (−1.11 − 1.11i)5-s + (−0.154 + 0.154i)7-s + 1.89i·9-s + (−1.05 + 1.05i)11-s + 1.29·13-s + 2.67i·15-s + (−0.532 − 0.846i)17-s + 1.72i·19-s + 0.372·21-s + (0.383 − 0.383i)23-s + 1.47i·25-s + (1.07 − 1.07i)27-s + (0.0364 + 0.0364i)29-s + (−0.931 − 0.931i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4708826803\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4708826803\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (2.19 + 3.48i)T \) |
good | 3 | \( 1 + (2.08 + 2.08i)T + 3iT^{2} \) |
| 5 | \( 1 + (2.48 + 2.48i)T + 5iT^{2} \) |
| 7 | \( 1 + (0.409 - 0.409i)T - 7iT^{2} \) |
| 11 | \( 1 + (3.51 - 3.51i)T - 11iT^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 19 | \( 1 - 7.51iT - 19T^{2} \) |
| 23 | \( 1 + (-1.83 + 1.83i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.196 - 0.196i)T + 29iT^{2} \) |
| 31 | \( 1 + (5.18 + 5.18i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.489 - 0.489i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - 41iT^{2} \) |
| 43 | \( 1 + 2.52iT - 43T^{2} \) |
| 47 | \( 1 + 1.63T + 47T^{2} \) |
| 53 | \( 1 + 2.97iT - 53T^{2} \) |
| 59 | \( 1 + 7.02iT - 59T^{2} \) |
| 61 | \( 1 + (-4.48 + 4.48i)T - 61iT^{2} \) |
| 67 | \( 1 - 0.490T + 67T^{2} \) |
| 71 | \( 1 + (-5.39 - 5.39i)T + 71iT^{2} \) |
| 73 | \( 1 + (-7.39 - 7.39i)T + 73iT^{2} \) |
| 79 | \( 1 + (9.35 - 9.35i)T - 79iT^{2} \) |
| 83 | \( 1 + 1.30iT - 83T^{2} \) |
| 89 | \( 1 + 0.335T + 89T^{2} \) |
| 97 | \( 1 + (-1.29 - 1.29i)T + 97iT^{2} \) |
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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
−9.972856286517722901638805866533, −8.764582863013259462983063131065, −7.927135966767944232482580749731, −7.48890286531223434341525278379, −6.49753370695727956137837728510, −5.56065410129207406033780937862, −4.89277111972686356451880634615, −3.85080470331262784335042917492, −2.02624233679042966451780327635, −0.828450748176073592224362159016,
0.36059726814914587188207996472, 3.07975410384749839787644248470, 3.67827789403478330289328332922, 4.61026955775156709167833809455, 5.61207366822922265943990851990, 6.39690284557615471186949063898, 7.18169416368543253535551888188, 8.338742464525185117557312134087, 9.103622439966464334451753873216, 10.40901099142211374274681795300