| L(s) = 1 | + i·2-s − 2·3-s + 4-s − 2i·6-s + 3i·8-s + 9-s + 2i·11-s − 2·12-s + (2 + 3i)13-s − 16-s + i·18-s + 6i·19-s − 2·22-s − 6·23-s − 6i·24-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.15·3-s + 0.5·4-s − 0.816i·6-s + 1.06i·8-s + 0.333·9-s + 0.603i·11-s − 0.577·12-s + (0.554 + 0.832i)13-s − 0.250·16-s + 0.235i·18-s + 1.37i·19-s − 0.426·22-s − 1.25·23-s − 1.22i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.436990 + 0.816524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436990 + 0.816524i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (-2 - 3i)T \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 2iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−11.99296353276332942642765155921, −11.01254498328097680870653599007, −10.37743657056284865208759440962, −9.022312494215980019091934159935, −7.83680075981377800263635606552, −6.93064912686883245100387620651, −6.02407049889489475068721471980, −5.44260138555209788819174274601, −4.01076134625495095606087851020, −1.93515407303545731145842992812,
0.75437596206238912134373358332, 2.60985512727022564354621993941, 3.96864114407920061231982211201, 5.51158480258020253772074522027, 6.19833588257915813134827660667, 7.25335192544233749893786306720, 8.531257305818010614689127322682, 9.882562489060455031237515278807, 10.65534114512796297854884259577, 11.42992459237653641999388807895
