L(s) = 1 | − 2-s + 4-s + 3·5-s − 2·7-s − 8-s − 3·10-s + 6·11-s + 2·14-s + 16-s + 3·17-s − 2·19-s + 3·20-s − 6·22-s + 6·23-s + 4·25-s − 2·28-s − 3·29-s + 4·31-s − 32-s − 3·34-s − 6·35-s + 7·37-s + 2·38-s − 3·40-s − 3·41-s − 10·43-s + 6·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 0.353·8-s − 0.948·10-s + 1.80·11-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.670·20-s − 1.27·22-s + 1.25·23-s + 4/5·25-s − 0.377·28-s − 0.557·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.01·35-s + 1.15·37-s + 0.324·38-s − 0.474·40-s − 0.468·41-s − 1.52·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.878436877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.878436877\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.978701072960327666773799396601, −8.146192139903195861150375917711, −6.99391053213761768135338329155, −6.49283772214867537360828430712, −6.00622210106381765069086775286, −5.03017618240304448996723918476, −3.79611665873254569331811277782, −2.91736744659519515633955578415, −1.83417296799279163989299255920, −0.994010259620585448539038750849,
0.994010259620585448539038750849, 1.83417296799279163989299255920, 2.91736744659519515633955578415, 3.79611665873254569331811277782, 5.03017618240304448996723918476, 6.00622210106381765069086775286, 6.49283772214867537360828430712, 6.99391053213761768135338329155, 8.146192139903195861150375917711, 8.978701072960327666773799396601