| L(s) = 1 | + 2.53·3-s + 3.24i·5-s + i·7-s + 3.43·9-s + 5.56i·11-s + (1.56 + 3.24i)13-s + 8.24i·15-s − 3.12·17-s − 6.56i·19-s + 2.53i·21-s + 0.712·23-s − 5.56·25-s + 1.11·27-s + 5.43·29-s − i·31-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.45i·5-s + 0.377i·7-s + 1.14·9-s + 1.67i·11-s + (0.433 + 0.901i)13-s + 2.12i·15-s − 0.757·17-s − 1.50i·19-s + 0.553i·21-s + 0.148·23-s − 1.11·25-s + 0.214·27-s + 1.00·29-s − 0.179i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.908436064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.908436064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-1.56 - 3.24i)T \) |
| good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 - 3.24iT - 5T^{2} \) |
| 11 | \( 1 - 5.56iT - 11T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 6.56iT - 19T^{2} \) |
| 23 | \( 1 - 0.712T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + iT - 31T^{2} \) |
| 37 | \( 1 + 3.96iT - 37T^{2} \) |
| 41 | \( 1 + 2.53iT - 41T^{2} \) |
| 43 | \( 1 + 8.32T + 43T^{2} \) |
| 47 | \( 1 - 12.3iT - 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 15.1iT - 59T^{2} \) |
| 61 | \( 1 + 0.438T + 61T^{2} \) |
| 67 | \( 1 - 0.684iT - 67T^{2} \) |
| 71 | \( 1 + 6.24iT - 71T^{2} \) |
| 73 | \( 1 - 4.36iT - 73T^{2} \) |
| 79 | \( 1 + 5.78T + 79T^{2} \) |
| 83 | \( 1 - 3.43iT - 83T^{2} \) |
| 89 | \( 1 - 3.24iT - 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−9.099174648845752820100788749680, −8.294260925037773003022807807062, −7.37988537238451368549496649007, −6.91631057109768729820638799195, −6.38684579258925969946961140194, −4.84770562036463343617975939818, −4.13202362628340645716404788559, −3.16300986965443370878252203453, −2.42834979379312375283624004187, −1.95782080608979883288479210764,
0.73426931225127635186087817402, 1.66678699457499472510570491811, 2.98514536262083547939295943461, 3.59987791150738233791428845361, 4.39854816495217984999610400013, 5.42426592346885675820673932877, 6.11173243377844602152282719737, 7.31785031573234665102542172409, 8.302518015025354977905207958528, 8.503656676832341839770428784772
