| L(s) = 1 | − 9i·5-s − 7·7-s − 9i·11-s + 14·13-s − 18i·17-s + 8·19-s + 36i·23-s − 56·25-s + 18i·29-s + 35·31-s + 63i·35-s + 44·37-s − 36i·41-s − 22·43-s + 54i·47-s + ⋯ |
| L(s) = 1 | − 1.80i·5-s − 7-s − 0.818i·11-s + 1.07·13-s − 1.05i·17-s + 0.421·19-s + 1.56i·23-s − 2.24·25-s + 0.620i·29-s + 1.12·31-s + 1.80i·35-s + 1.18·37-s − 0.878i·41-s − 0.511·43-s + 1.14i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.826166 - 0.826166i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.826166 - 0.826166i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9iT - 25T^{2} \) |
| 7 | \( 1 + 7T + 49T^{2} \) |
| 11 | \( 1 + 9iT - 121T^{2} \) |
| 13 | \( 1 - 14T + 169T^{2} \) |
| 17 | \( 1 + 18iT - 289T^{2} \) |
| 19 | \( 1 - 8T + 361T^{2} \) |
| 23 | \( 1 - 36iT - 529T^{2} \) |
| 29 | \( 1 - 18iT - 841T^{2} \) |
| 31 | \( 1 - 35T + 961T^{2} \) |
| 37 | \( 1 - 44T + 1.36e3T^{2} \) |
| 41 | \( 1 + 36iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22T + 1.84e3T^{2} \) |
| 47 | \( 1 - 54iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 18iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20T + 3.72e3T^{2} \) |
| 67 | \( 1 - 14T + 4.48e3T^{2} \) |
| 71 | \( 1 + 126iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 89T + 5.32e3T^{2} \) |
| 79 | \( 1 - 110T + 6.24e3T^{2} \) |
| 83 | \( 1 + 27iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 18iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 11T + 9.40e3T^{2} \) |
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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
−13.32400762138401490995873481208, −12.27763000891051257122473646923, −11.30700392146741847634093170779, −9.626863263157341712288610308841, −9.010188945624815320013604632274, −7.903536057022504246604496228115, −6.16503019899032015256015540590, −5.07357916438979434651728887417, −3.51759935779301444893503367775, −0.920305415319379111537155552069,
2.64442532676650251209430376905, 3.89472861492659457570043490807, 6.23816933355328191565619285740, 6.72210216652094354861510849865, 8.131754965503335801945351869534, 9.812943568856192976160582953005, 10.44805001704607546789308730683, 11.47249956539311320157921856633, 12.77064333225368885393886034155, 13.77755173805365306314427564469
