| L(s) = 1 | + (−1.58 − 0.707i)3-s + 4.24i·7-s + (2.00 + 2.23i)9-s + (3 − 6.70i)21-s − 9.48·23-s + (−1.58 − 4.94i)27-s − 8.94i·29-s + 4.47i·41-s + 12.7i·43-s − 9.48·47-s − 10.9·49-s − 8·61-s + (−9.48 + 8.48i)63-s − 4.24i·67-s + (15.0 + 6.70i)69-s + ⋯ |
| L(s) = 1 | + (−0.912 − 0.408i)3-s + 1.60i·7-s + (0.666 + 0.745i)9-s + (0.654 − 1.46i)21-s − 1.97·23-s + (−0.304 − 0.952i)27-s − 1.66i·29-s + 0.698i·41-s + 1.94i·43-s − 1.38·47-s − 1.57·49-s − 1.02·61-s + (−1.19 + 1.06i)63-s − 0.518i·67-s + (1.80 + 0.807i)69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3782701749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3782701749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.58 + 0.707i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 9.48T + 23T^{2} \) |
| 29 | \( 1 + 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.979504296508824130134611003035, −9.477047769601155711751232519987, −8.231006165647596421600248845937, −7.82275182568216647427483607342, −6.36182810065420050000545529189, −6.08731984419186035290178581779, −5.19745834807429971860895813244, −4.24921464296036442236719590495, −2.68445029283898720053637630623, −1.72726608917248872219843368270,
0.18175536164629201967457014113, 1.55974140955655215379962410735, 3.52403684896991389277666314904, 4.15542377000874807303169106813, 5.05688802381976880694073409835, 6.03634019887939705435999547990, 6.92509261156463247815173193532, 7.50150924273585404240967450970, 8.613231900772013984016295079343, 9.722626701631854934566639801314
