L(s) = 1 | + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.707 + 0.292i)11-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (0.707 − 0.292i)29-s + (−0.707 + 1.70i)37-s + (1.70 + 0.707i)43-s + 1.00i·49-s + (−1.70 − 0.707i)53-s − 1.00·63-s + (−1.70 + 0.707i)67-s + (−0.292 − 0.707i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (0.707 + 0.292i)11-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (0.707 − 0.292i)29-s + (−0.707 + 1.70i)37-s + (1.70 + 0.707i)43-s + 1.00i·49-s + (−1.70 − 0.707i)53-s − 1.00·63-s + (−1.70 + 0.707i)67-s + (−0.292 − 0.707i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001402116\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001402116\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.07714143531296779856097997375, −9.578036305130516829698202428167, −8.672870259907173974021927159507, −7.57494709329354829425291154387, −6.66499596100028099353452090499, −6.28856204829249826704379628031, −4.68245350256690288926292139439, −3.98667709605783841911901767470, −2.90114223147121109838553647662, −1.17278319926453268689170926646,
1.68642899387736928874792203941, 3.01838524610478155287104693419, 4.04899665595233899823002489118, 5.23372999870034590157920533037, 6.03269040468560780819678370674, 7.05160542580676998808925786043, 7.75023948533664506084751229972, 9.067152772628851598352349845602, 9.301182373677471663919291295137, 10.43114020442701134391521099516