L(s) = 1 | + (−1.10 + 1.59i)2-s + (−0.982 − 2.59i)4-s + (0.568 − 0.822i)7-s + (3.33 + 0.822i)8-s + (0.120 − 0.992i)9-s + (0.213 + 0.112i)11-s + (0.688 + 1.81i)14-s + (−2.92 + 2.59i)16-s + (1.45 + 1.28i)18-s + (−0.414 + 0.217i)22-s + 1.77·23-s + (−0.970 − 0.239i)25-s + (−2.69 − 0.663i)28-s + (−1.71 + 0.902i)29-s + (−0.499 − 4.11i)32-s + ⋯ |
L(s) = 1 | + (−1.10 + 1.59i)2-s + (−0.982 − 2.59i)4-s + (0.568 − 0.822i)7-s + (3.33 + 0.822i)8-s + (0.120 − 0.992i)9-s + (0.213 + 0.112i)11-s + (0.688 + 1.81i)14-s + (−2.92 + 2.59i)16-s + (1.45 + 1.28i)18-s + (−0.414 + 0.217i)22-s + 1.77·23-s + (−0.970 − 0.239i)25-s + (−2.69 − 0.663i)28-s + (−1.71 + 0.902i)29-s + (−0.499 − 4.11i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4908761843\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4908761843\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.568 + 0.822i)T \) |
| 53 | \( 1 + (-0.120 - 0.992i)T \) |
good | 2 | \( 1 + (1.10 - 1.59i)T + (-0.354 - 0.935i)T^{2} \) |
| 3 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 5 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 11 | \( 1 + (-0.213 - 0.112i)T + (0.568 + 0.822i)T^{2} \) |
| 13 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 17 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 19 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 23 | \( 1 - 1.77T + T^{2} \) |
| 29 | \( 1 + (1.71 - 0.902i)T + (0.568 - 0.822i)T^{2} \) |
| 31 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 37 | \( 1 + (-0.530 + 0.470i)T + (0.120 - 0.992i)T^{2} \) |
| 41 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 43 | \( 1 + (-1.12 - 0.992i)T + (0.120 + 0.992i)T^{2} \) |
| 47 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 59 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 61 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 67 | \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \) |
| 71 | \( 1 + (0.850 + 0.753i)T + (0.120 + 0.992i)T^{2} \) |
| 73 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 79 | \( 1 + (-0.645 - 0.935i)T + (-0.354 + 0.935i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 97 | \( 1 + (0.970 + 0.239i)T^{2} \) |
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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
−11.29528365837766818739156441989, −10.55762236637817263131102664968, −9.435181760754342608736789015246, −9.027749725414668082857967808798, −7.78768675474191657690433811374, −7.21574527615966156620471333891, −6.33731376380385662283226435764, −5.28144281919149885574651713113, −4.10882263662856725984741214040, −1.21483920791398874146588636035,
1.72824744916573519901804430076, 2.70625087826847940274429936770, 4.09692696163911337333726664493, 5.33119413206694661542310281635, 7.36239858282369892172752074168, 8.107941921774576925025561498421, 8.986246644160028144693245340541, 9.653953175467028974332683173150, 10.77085159991824567933217325619, 11.29035070441712563717621524381