L(s) = 1 | + (−0.984 − 0.984i)2-s + (−1.25 − 0.724i)3-s − 0.0619i·4-s + (−0.172 − 0.643i)5-s + (0.521 + 1.94i)6-s + (−2.46 + 0.960i)7-s + (−2.02 + 2.02i)8-s + (−0.450 − 0.780i)9-s + (−0.463 + 0.802i)10-s + (−1.24 − 4.65i)11-s + (−0.0448 + 0.0776i)12-s + (3.60 + 0.0282i)13-s + (3.37 + 1.48i)14-s + (−0.249 + 0.931i)15-s + 3.87·16-s + 0.467·17-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.696i)2-s + (−0.724 − 0.418i)3-s − 0.0309i·4-s + (−0.0770 − 0.287i)5-s + (0.213 + 0.795i)6-s + (−0.931 + 0.363i)7-s + (−0.717 + 0.717i)8-s + (−0.150 − 0.260i)9-s + (−0.146 + 0.253i)10-s + (−0.376 − 1.40i)11-s + (−0.0129 + 0.0224i)12-s + (0.999 + 0.00782i)13-s + (0.901 + 0.395i)14-s + (−0.0644 + 0.240i)15-s + 0.968·16-s + 0.113·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0564705 - 0.407171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0564705 - 0.407171i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.46 - 0.960i)T \) |
| 13 | \( 1 + (-3.60 - 0.0282i)T \) |
good | 2 | \( 1 + (0.984 + 0.984i)T + 2iT^{2} \) |
| 3 | \( 1 + (1.25 + 0.724i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.172 + 0.643i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (1.24 + 4.65i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 - 0.467T + 17T^{2} \) |
| 19 | \( 1 + (-3.26 - 0.873i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 6.95iT - 23T^{2} \) |
| 29 | \( 1 + (2.01 + 3.49i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.10 + 1.09i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.38 - 2.38i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.68 - 0.986i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.42 - 1.97i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.64 - 2.58i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.20 - 3.81i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.33 - 4.33i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.21 + 2.43i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.03 - 2.42i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.19 + 0.857i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (0.0301 - 0.112i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-0.194 + 0.337i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.83 - 6.83i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.61 + 17.2i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−13.27768266245316109064076270772, −12.27317777582661924091413132323, −11.35514876347539828806416954029, −10.54594243528466436964729771954, −9.228507683102188813100261763998, −8.407620697697599736266317667657, −6.34761955190753202711115555274, −5.66603505115740834464194385750, −3.11293225848952920042590814833, −0.68484255064352427069111382166,
3.55422060863412940283843377287, 5.42480643743911343545089810350, 6.79772597347719281572412439298, 7.62042038146658830272843520439, 9.179049947770543641124921342946, 10.07191477716812111318314517019, 11.15993416931236022446562656214, 12.45417817395701758772838411414, 13.42657467558342557037786585502, 15.03279395651624849852675080445