| L(s) = 1 | + 3.41·5-s + 4.82·11-s − 1.41·13-s − 6.24·17-s + 1.17·19-s − 0.828·23-s + 6.65·25-s + 8.48·29-s + 10.8·31-s − 9.65·37-s − 3.41·41-s + 8·43-s − 1.17·47-s − 9.31·53-s + 16.4·55-s − 10.8·59-s + 5.89·61-s − 4.82·65-s + 8·67-s + 4.82·71-s − 3.07·73-s + 13.6·79-s + 7.31·83-s − 21.3·85-s + 14.7·89-s + 4·95-s + 16.2·97-s + ⋯ |
| L(s) = 1 | + 1.52·5-s + 1.45·11-s − 0.392·13-s − 1.51·17-s + 0.268·19-s − 0.172·23-s + 1.33·25-s + 1.57·29-s + 1.94·31-s − 1.58·37-s − 0.533·41-s + 1.21·43-s − 0.170·47-s − 1.27·53-s + 2.22·55-s − 1.40·59-s + 0.755·61-s − 0.598·65-s + 0.977·67-s + 0.573·71-s − 0.359·73-s + 1.53·79-s + 0.802·83-s − 2.31·85-s + 1.56·89-s + 0.410·95-s + 1.64·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.102951183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.102951183\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 4.82T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.050924452631432046499742664566, −6.90850099247763940937533884014, −6.42798757349282460207249498004, −6.14912847357666000642035102585, −4.96836802195540302830193397001, −4.60435409424163839205754615601, −3.50870688170774743165502972582, −2.51579867666102937448321873435, −1.87153516808192244056488864025, −0.935231669727870454811511582386,
0.935231669727870454811511582386, 1.87153516808192244056488864025, 2.51579867666102937448321873435, 3.50870688170774743165502972582, 4.60435409424163839205754615601, 4.96836802195540302830193397001, 6.14912847357666000642035102585, 6.42798757349282460207249498004, 6.90850099247763940937533884014, 8.050924452631432046499742664566
