| L(s) = 1 | + 2i·5-s − 7-s − 2i·11-s + 6i·13-s + 6·17-s − 8i·19-s + 8·23-s + 25-s − 8·31-s − 2i·35-s − 4i·37-s − 6·41-s + 6i·43-s + 12·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s − 0.377·7-s − 0.603i·11-s + 1.66i·13-s + 1.45·17-s − 1.83i·19-s + 1.66·23-s + 0.200·25-s − 1.43·31-s − 0.338i·35-s − 0.657i·37-s − 0.937·41-s + 0.914i·43-s + 1.75·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898832762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898832762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.808403309351388672536331540702, −7.55160282233112784942132997330, −6.94320062609902561388226159792, −6.61921990279707123021578868475, −5.56526152050135321541992395755, −4.84237008492036532612821226815, −3.74506568742597022604786064860, −3.10194262993791376912793064246, −2.27268346840860367279736915015, −0.909230652591034306570431116367,
0.73827499140092141964850160956, 1.62193942419718725764356837302, 3.04970745004688168731530314786, 3.58840272941922389825174386599, 4.70340125006679818102810707053, 5.54885722870958506325415901972, 5.75024090520893590247927042600, 7.09630123037333113588879970789, 7.64681512079672486673555495127, 8.337934849524808270569581995651
