| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 13-s − 14-s + 16-s + 3·17-s + 2·19-s + 3·23-s + 26-s − 28-s − 3·29-s − 31-s + 32-s + 3·34-s − 2·37-s + 2·38-s + 3·41-s + 7·43-s + 3·46-s + 6·47-s + 49-s + 52-s + 9·53-s − 56-s − 3·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s + 0.625·23-s + 0.196·26-s − 0.188·28-s − 0.557·29-s − 0.179·31-s + 0.176·32-s + 0.514·34-s − 0.328·37-s + 0.324·38-s + 0.468·41-s + 1.06·43-s + 0.442·46-s + 0.875·47-s + 1/7·49-s + 0.138·52-s + 1.23·53-s − 0.133·56-s − 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.006680894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.006680894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.745857990813457817329955102093, −7.65077274986995613409960280936, −7.22951560110957571435294807910, −6.23633129866797194691812367250, −5.65204380484628468493197445604, −4.86914755909323308164532308187, −3.88697005176365350523651137986, −3.23172026946449828635329767073, −2.26147685347986875711559282796, −0.973743749877719920965802819698,
0.973743749877719920965802819698, 2.26147685347986875711559282796, 3.23172026946449828635329767073, 3.88697005176365350523651137986, 4.86914755909323308164532308187, 5.65204380484628468493197445604, 6.23633129866797194691812367250, 7.22951560110957571435294807910, 7.65077274986995613409960280936, 8.745857990813457817329955102093
