| L(s) = 1 | + 3.30i·5-s − 3.45·7-s + 3.09i·11-s + 2.50i·13-s + 17-s + 4.43i·19-s + 4.14·23-s − 5.91·25-s + 7.15i·29-s + 6.22·31-s − 11.4i·35-s + 6.74i·37-s + 6.99·41-s + 2.22i·43-s − 3.31·47-s + ⋯ |
| L(s) = 1 | + 1.47i·5-s − 1.30·7-s + 0.933i·11-s + 0.695i·13-s + 0.242·17-s + 1.01i·19-s + 0.863·23-s − 1.18·25-s + 1.32i·29-s + 1.11·31-s − 1.93i·35-s + 1.10i·37-s + 1.09·41-s + 0.339i·43-s − 0.483·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184052998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184052998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3.30iT - 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 - 3.09iT - 11T^{2} \) |
| 13 | \( 1 - 2.50iT - 13T^{2} \) |
| 19 | \( 1 - 4.43iT - 19T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 - 7.15iT - 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 - 6.74iT - 37T^{2} \) |
| 41 | \( 1 - 6.99T + 41T^{2} \) |
| 43 | \( 1 - 2.22iT - 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 - 6.19iT - 53T^{2} \) |
| 59 | \( 1 - 8.83iT - 59T^{2} \) |
| 61 | \( 1 + 14.3iT - 61T^{2} \) |
| 67 | \( 1 + 6.44iT - 67T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 7.32T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
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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
−8.701952568293476671824716635379, −7.62940158783116095228867976887, −7.11241443830025121602441463320, −6.47749548780418669002052651927, −6.12330901408682638517735994455, −4.93131391005768430576567724258, −3.99688850313251677979741468147, −3.14690894758259274277311516041, −2.72545586711703849357659006440, −1.51387595940245869002779926317,
0.39331368797045175051529907256, 0.946489385413362508158542465192, 2.52791866320673981553837903756, 3.27425528740878320144815806548, 4.18569225281311856908016000677, 4.96915147888159372588626782688, 5.75560105520396041571464828350, 6.23741808514127896005136461108, 7.21686460237211639358054515899, 8.006805246585816108557249852457
