| L(s) = 1 | − i·3-s − 9-s + 4·11-s − 6i·13-s + 6i·17-s − 4·19-s + i·27-s − 2·29-s − 8·31-s − 4i·33-s − 2i·37-s − 6·39-s − 6·41-s − 12i·43-s − 8i·47-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.333·9-s + 1.20·11-s − 1.66i·13-s + 1.45i·17-s − 0.917·19-s + 0.192i·27-s − 0.371·29-s − 1.43·31-s − 0.696i·33-s − 0.328i·37-s − 0.960·39-s − 0.937·41-s − 1.82i·43-s − 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079077209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079077209\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.088078085062497659881020821369, −7.16042493093439203673160904572, −6.61548065550678293916053145047, −5.76204000281470405600526617975, −5.31309530311752212479735715925, −3.89857759123985928834040086071, −3.60445435903543290321774038593, −2.29414568521590372842859477024, −1.50794429003346791380644018311, −0.28990047752366773298050040821,
1.35172418208740981986415162107, 2.33968411710848432369505541757, 3.40882284369630044140144217101, 4.24464911594140542575326212724, 4.65328193659855710671160174356, 5.66910998952363233241607107513, 6.55491585043535998823067326448, 6.95733613986660732011496397802, 7.88172147093439417105074732699, 8.918197958112961084216394547938
