| L(s) = 1 | + 2i·7-s − 1.41·11-s − 13-s + 1.41i·17-s + 4i·19-s + 2.82·23-s + 5·25-s + 9.89i·29-s + 2·37-s − 2.82i·41-s − 4i·43-s − 9.89·47-s + 3·49-s − 4.24i·53-s − 1.41·59-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s − 0.426·11-s − 0.277·13-s + 0.342i·17-s + 0.917i·19-s + 0.589·23-s + 25-s + 1.83i·29-s + 0.328·37-s − 0.441i·41-s − 0.609i·43-s − 1.44·47-s + 0.428·49-s − 0.582i·53-s − 0.184·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076568395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076568395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2.82iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.259653709485818782199536262385, −7.47301798124843684061664161442, −6.77456470613577402512093422331, −6.08283342615326817445650242079, −5.22418583352148922284503006884, −4.90695935661309983518185932572, −3.72084106637633549003127158598, −3.05410932742519117314909649907, −2.19072989636188566162693606694, −1.25214381098525281955166205498,
0.27150572431376904006040150794, 1.24019311921885996675135745622, 2.52205122614482167652039647396, 3.06215217720542047608658397282, 4.18515884012541841582530939555, 4.68579080409422515425999368786, 5.42539961956459633495941206236, 6.36475446483165807490649111671, 6.93541139854434095665543617136, 7.61280283418713411662956862006
