| L(s) = 1 | − 3-s + 2.44·5-s + 7-s + 9-s − 4.44·11-s + 2.89·13-s − 2.44·15-s + 6.44·17-s − 2.89·19-s − 21-s + 0.449·23-s + 0.999·25-s − 27-s + 2.89·29-s + 6.89·31-s + 4.44·33-s + 2.44·35-s + 2·37-s − 2.89·39-s − 7.34·41-s − 3.10·43-s + 2.44·45-s − 0.898·47-s + 49-s − 6.44·51-s + 10·53-s − 10.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.09·5-s + 0.377·7-s + 0.333·9-s − 1.34·11-s + 0.804·13-s − 0.632·15-s + 1.56·17-s − 0.665·19-s − 0.218·21-s + 0.0937·23-s + 0.199·25-s − 0.192·27-s + 0.538·29-s + 1.23·31-s + 0.774·33-s + 0.414·35-s + 0.328·37-s − 0.464·39-s − 1.14·41-s − 0.472·43-s + 0.365·45-s − 0.131·47-s + 0.142·49-s − 0.903·51-s + 1.37·53-s − 1.46·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.124147410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124147410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 0.449T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 - 6.89T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + 0.898T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 - 5.10T + 97T^{2} \) |
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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
−8.125729616692856629127428231228, −7.55007030563806169005130121214, −6.46393892779863554127241669718, −6.03793394555653502998608530406, −5.26105147032306915020766840440, −4.87035716802883538706812606349, −3.67581315931513537494532436974, −2.70947110218877003576729846505, −1.81725476195762217407518451970, −0.834368080654622395164969259254,
0.834368080654622395164969259254, 1.81725476195762217407518451970, 2.70947110218877003576729846505, 3.67581315931513537494532436974, 4.87035716802883538706812606349, 5.26105147032306915020766840440, 6.03793394555653502998608530406, 6.46393892779863554127241669718, 7.55007030563806169005130121214, 8.125729616692856629127428231228
