| L(s) = 1 | − 3-s + 1.04·5-s − 4.50·7-s + 9-s − 1.23·11-s − 6.12·13-s − 1.04·15-s − 3.97·17-s − 4.32·19-s + 4.50·21-s − 0.706·23-s − 3.91·25-s − 27-s − 4.57·29-s − 5.21·31-s + 1.23·33-s − 4.68·35-s − 8.53·37-s + 6.12·39-s − 9.49·41-s + 3.33·43-s + 1.04·45-s + 0.839·47-s + 13.3·49-s + 3.97·51-s + 8.83·53-s − 1.28·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.465·5-s − 1.70·7-s + 0.333·9-s − 0.371·11-s − 1.69·13-s − 0.268·15-s − 0.963·17-s − 0.991·19-s + 0.983·21-s − 0.147·23-s − 0.783·25-s − 0.192·27-s − 0.849·29-s − 0.936·31-s + 0.214·33-s − 0.792·35-s − 1.40·37-s + 0.980·39-s − 1.48·41-s + 0.508·43-s + 0.155·45-s + 0.122·47-s + 1.90·49-s + 0.556·51-s + 1.21·53-s − 0.172·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08816032962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08816032962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 + 3.97T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 + 0.706T + 23T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 + 8.53T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 - 3.33T + 43T^{2} \) |
| 47 | \( 1 - 0.839T + 47T^{2} \) |
| 53 | \( 1 - 8.83T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 2.04T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 8.45T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 8.74T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.58005359103969678702410764146, −6.88655297496354778028089902969, −6.62633867603231030718278180597, −5.66519094661935127190885001500, −5.29035200521015565372536074495, −4.26451018702025591174611011655, −3.56830042587467960660392061062, −2.51696238575639443977404175826, −1.98370660883590060897780355504, −0.13800697998428567837679746669,
0.13800697998428567837679746669, 1.98370660883590060897780355504, 2.51696238575639443977404175826, 3.56830042587467960660392061062, 4.26451018702025591174611011655, 5.29035200521015565372536074495, 5.66519094661935127190885001500, 6.62633867603231030718278180597, 6.88655297496354778028089902969, 7.58005359103969678702410764146
