L(s) = 1 | + (0.707 + 0.707i)3-s + (1.72 − 1.42i)5-s + (1.66 + 1.66i)7-s + 1.00i·9-s + 0.0251i·11-s + (4.31 + 4.31i)13-s + (2.22 + 0.210i)15-s + (−2.93 − 2.93i)17-s + 5.73·19-s + 2.35i·21-s + (−4.49 − 1.66i)23-s + (0.938 − 4.91i)25-s + (−0.707 + 0.707i)27-s + 7.21i·29-s − 8.64·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (0.770 − 0.637i)5-s + (0.630 + 0.630i)7-s + 0.333i·9-s + 0.00757i·11-s + (1.19 + 1.19i)13-s + (0.574 + 0.0544i)15-s + (−0.711 − 0.711i)17-s + 1.31·19-s + 0.514i·21-s + (−0.938 − 0.346i)23-s + (0.187 − 0.982i)25-s + (−0.136 + 0.136i)27-s + 1.34i·29-s − 1.55·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.489229081\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.489229081\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.72 + 1.42i)T \) |
| 23 | \( 1 + (4.49 + 1.66i)T \) |
good | 7 | \( 1 + (-1.66 - 1.66i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.0251iT - 11T^{2} \) |
| 13 | \( 1 + (-4.31 - 4.31i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.93 + 2.93i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 29 | \( 1 - 7.21iT - 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 + (-6.47 - 6.47i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.0620T + 41T^{2} \) |
| 43 | \( 1 + (-5.43 + 5.43i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.36 - 4.36i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.27 + 9.27i)T - 53iT^{2} \) |
| 59 | \( 1 - 2.10iT - 59T^{2} \) |
| 61 | \( 1 + 4.81iT - 61T^{2} \) |
| 67 | \( 1 + (0.147 + 0.147i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 + (1.58 + 1.58i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 + (0.182 - 0.182i)T - 83iT^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + (-12.8 - 12.8i)T + 97iT^{2} \) |
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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
−9.386924595671451946133648952371, −8.940441701482973409823348718994, −8.388073277002800021145816775112, −7.25955208286519874108182717536, −6.23016514234166189859709978568, −5.36444264007773985038922182290, −4.67764320763819581991312577786, −3.65068310292515267246080424760, −2.32511471155904374469172852282, −1.45070791609846784652730985344,
1.13649043297460322128786430104, 2.20525579888258194686521158055, 3.33457618309566676372983597708, 4.19310045964558037249647005462, 5.74917275670992924308612497246, 5.99131060737764730940947480516, 7.31830593051754411877529878094, 7.72388606529589031112997853664, 8.642141263370761672971836790475, 9.536028320625579394649756098179