| L(s) = 1 | − 2.44·3-s − 2.44·7-s + 2.99·9-s − 4.89·11-s − 4·17-s + 4.89·19-s + 5.99·21-s − 2.44·23-s + 8·29-s − 9.79·31-s + 11.9·33-s − 4·37-s − 8·41-s − 7.34·43-s − 12.2·47-s − 1.00·49-s + 9.79·51-s − 8·53-s − 11.9·57-s − 4.89·59-s + 6·61-s − 7.34·63-s − 2.44·67-s + 5.99·69-s + 9.79·71-s − 4·73-s + 11.9·77-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 0.925·7-s + 0.999·9-s − 1.47·11-s − 0.970·17-s + 1.12·19-s + 1.30·21-s − 0.510·23-s + 1.48·29-s − 1.75·31-s + 2.08·33-s − 0.657·37-s − 1.24·41-s − 1.12·43-s − 1.78·47-s − 0.142·49-s + 1.37·51-s − 1.09·53-s − 1.58·57-s − 0.637·59-s + 0.768·61-s − 0.925·63-s − 0.299·67-s + 0.722·69-s + 1.16·71-s − 0.468·73-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1746929808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1746929808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 9.79T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 7.34T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 9.79T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−7.938712579674749341027435627717, −7.04640742334732456526016431617, −6.56536775431522621880388433485, −5.91642868261680090152416654750, −5.05492272558939153819359159147, −4.90106940327372207356123000447, −3.56119392165859084201390689793, −2.86439867737768881148439064396, −1.66680322850065496209100505128, −0.22620076627417607289201011686,
0.22620076627417607289201011686, 1.66680322850065496209100505128, 2.86439867737768881148439064396, 3.56119392165859084201390689793, 4.90106940327372207356123000447, 5.05492272558939153819359159147, 5.91642868261680090152416654750, 6.56536775431522621880388433485, 7.04640742334732456526016431617, 7.938712579674749341027435627717
