| L(s) = 1 | − i·3-s + 2.60i·7-s − 9-s − 4.60·11-s + 4.60i·13-s − 2i·17-s + 19-s + 2.60·21-s − 2i·23-s + i·27-s − 2.60·29-s + 4·31-s + 4.60i·33-s − 3.39i·37-s + 4.60·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.984i·7-s − 0.333·9-s − 1.38·11-s + 1.27i·13-s − 0.485i·17-s + 0.229·19-s + 0.568·21-s − 0.417i·23-s + 0.192i·27-s − 0.483·29-s + 0.718·31-s + 0.801i·33-s − 0.558i·37-s + 0.737·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2648389751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2648389751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2.60iT - 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 - 4.60iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 3.39iT - 37T^{2} \) |
| 41 | \( 1 - 6.60T + 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 6.60T + 89T^{2} \) |
| 97 | \( 1 + 16.6iT - 97T^{2} \) |
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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
−7.73518557757383096444520941345, −7.23505407028710860275505878620, −6.32234383937065644244392307347, −5.77025204884411839835014658840, −4.99702400868419346451952731497, −4.27501020112365161287791619108, −2.93988684583570338081750760846, −2.48290764190058555014588449912, −1.56307905192353597391027656364, −0.07249247824167750068380562156,
1.07064095967041175836559923059, 2.50565015676668196663289824905, 3.22679198371230744530716961061, 4.01019156313983760904619432840, 4.81721149186834413341321997459, 5.50101873928345931203042596592, 6.10498661769969313597509720724, 7.25688612780065685732040137685, 7.72954949473979203474732696153, 8.273433856473623924519764163337
