L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + 1.00i·15-s + (1.41 − 1.41i)17-s − 1.41·19-s − 1.00·21-s + (−1 + i)23-s − 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·31-s + 1.00·35-s + (1 − i)37-s − 1.41·41-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + 1.00i·15-s + (1.41 − 1.41i)17-s − 1.41·19-s − 1.00·21-s + (−1 + i)23-s − 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·31-s + 1.00·35-s + (1 − i)37-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9738653440\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9738653440\) |
\(L(1)\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.284588015155818561483837380060, −7.62887699043896466917023439513, −7.31007780879409605649140581558, −6.47422340273375297653353459868, −5.83101266171108791195686761391, −4.37651429734154422247156624427, −3.63189351152470652678099765184, −3.04569477009691955604367886586, −2.06096572333113468675196477700, −0.51267878725346465465903439096,
1.66989948474539997603693503365, 2.81701798163101577554049097297, 3.65805592110341863055821231420, 4.25121629032857723869808032716, 5.14125631528036360454889034407, 5.95129184767025811922896185261, 6.80896370279099142041000967867, 8.035267091361205260211748305195, 8.382692510544872679309752573399, 8.788480309901355348589718034493