L(s) = 1 | + 7-s + 2·13-s − 6·17-s − 4·19-s − 6·29-s + 4·31-s + 2·37-s − 6·41-s − 8·43-s + 12·47-s + 49-s − 6·53-s + 12·59-s − 2·61-s − 8·67-s − 14·73-s + 16·79-s + 12·83-s − 6·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.977·67-s − 1.63·73-s + 1.80·79-s + 1.31·83-s − 0.635·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.453600793\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.453600793\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.60882408495419, −13.32264728757848, −12.91884075551112, −12.20933450102865, −11.78451214845615, −11.23627378199644, −10.83723954264232, −10.43758932912401, −9.827937629171113, −9.089479453688010, −8.883319464817576, −8.221795269581910, −7.916339952047820, −7.042447592159100, −6.745511480102752, −6.164012110169864, −5.602011943408022, −4.986843203894144, −4.344049111490761, −4.020932719033894, −3.282125250649896, −2.508526379982000, −1.986532181408582, −1.357754927690254, −0.3726614540654584,
0.3726614540654584, 1.357754927690254, 1.986532181408582, 2.508526379982000, 3.282125250649896, 4.020932719033894, 4.344049111490761, 4.986843203894144, 5.602011943408022, 6.164012110169864, 6.745511480102752, 7.042447592159100, 7.916339952047820, 8.221795269581910, 8.883319464817576, 9.089479453688010, 9.827937629171113, 10.43758932912401, 10.83723954264232, 11.23627378199644, 11.78451214845615, 12.20933450102865, 12.91884075551112, 13.32264728757848, 13.60882408495419