| L(s) = 1 | − 2.44·2-s + 3.99·4-s + 7-s − 4.89·8-s − 2.44·11-s − 4.44·13-s − 2.44·14-s + 3.99·16-s − 5.44·17-s + 4.44·19-s + 5.99·22-s − 23-s + 10.8·26-s + 3.99·28-s − 10.3·29-s + 0.101·31-s + 13.3·34-s − 3.89·37-s − 10.8·38-s − 5.44·41-s − 2·43-s − 9.79·44-s + 2.44·46-s + 8.44·47-s − 6·49-s − 17.7·52-s + 0.550·53-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 1.99·4-s + 0.377·7-s − 1.73·8-s − 0.738·11-s − 1.23·13-s − 0.654·14-s + 0.999·16-s − 1.32·17-s + 1.02·19-s + 1.27·22-s − 0.208·23-s + 2.13·26-s + 0.755·28-s − 1.92·29-s + 0.0181·31-s + 2.28·34-s − 0.640·37-s − 1.76·38-s − 0.851·41-s − 0.304·43-s − 1.47·44-s + 0.361·46-s + 1.23·47-s − 0.857·49-s − 2.46·52-s + 0.0756·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4213705490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4213705490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.101T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 8.44T + 47T^{2} \) |
| 53 | \( 1 - 0.550T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 0.651T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 - 4.34T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−8.183823964703705345490648691676, −7.60271416611506680756748893992, −7.20390779281586171831704446390, −6.40364831111884222109081333860, −5.36716719939435300510940687632, −4.70474211835085977013270317960, −3.40012569963787382033760878765, −2.31628291613187818827673578168, −1.83688796610594970730729068055, −0.43707490691601148699989059465,
0.43707490691601148699989059465, 1.83688796610594970730729068055, 2.31628291613187818827673578168, 3.40012569963787382033760878765, 4.70474211835085977013270317960, 5.36716719939435300510940687632, 6.40364831111884222109081333860, 7.20390779281586171831704446390, 7.60271416611506680756748893992, 8.183823964703705345490648691676
