| L(s) = 1 | + 1.54·2-s − 2.43·3-s + 0.393·4-s − 3.76·6-s + 2.19·7-s − 2.48·8-s + 2.92·9-s − 1.23·11-s − 0.958·12-s + 4.52·13-s + 3.40·14-s − 4.63·16-s − 1.39·17-s + 4.52·18-s − 3.20·19-s − 5.35·21-s − 1.91·22-s + 6.33·23-s + 6.04·24-s + 7.00·26-s + 0.185·27-s + 0.865·28-s + 3.14·29-s + 5.35·31-s − 2.19·32-s + 3.00·33-s − 2.16·34-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 1.40·3-s + 0.196·4-s − 1.53·6-s + 0.830·7-s − 0.878·8-s + 0.974·9-s − 0.372·11-s − 0.276·12-s + 1.25·13-s + 0.909·14-s − 1.15·16-s − 0.338·17-s + 1.06·18-s − 0.735·19-s − 1.16·21-s − 0.407·22-s + 1.32·23-s + 1.23·24-s + 1.37·26-s + 0.0357·27-s + 0.163·28-s + 0.584·29-s + 0.961·31-s − 0.388·32-s + 0.523·33-s − 0.370·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659895974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659895974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 7 | \( 1 - 2.19T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 - 5.35T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 47 | \( 1 - 5.92T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 + 9.94T + 67T^{2} \) |
| 71 | \( 1 + 7.02T + 71T^{2} \) |
| 73 | \( 1 + 1.33T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 9.96T + 97T^{2} \) |
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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
−10.29042094729893301025116769742, −8.963931639188851903460735872449, −8.260802990287052061801434037664, −6.91618163506218649900595338809, −6.14184514830362615755641025322, −5.55001013133355311445559579716, −4.70653933121933412198994960898, −4.15948481214430979093667481540, −2.72687959298251747220691739430, −0.932421443809388546744567600525,
0.932421443809388546744567600525, 2.72687959298251747220691739430, 4.15948481214430979093667481540, 4.70653933121933412198994960898, 5.55001013133355311445559579716, 6.14184514830362615755641025322, 6.91618163506218649900595338809, 8.260802990287052061801434037664, 8.963931639188851903460735872449, 10.29042094729893301025116769742
