| L(s) = 1 | − 5-s + 2·7-s + 4·11-s + 13-s + 4·17-s − 6·19-s + 25-s − 4·29-s + 6·31-s − 2·35-s + 2·37-s − 10·41-s − 8·43-s − 3·49-s − 4·53-s − 4·55-s + 4·59-s − 2·61-s − 65-s − 6·67-s + 8·71-s + 10·73-s + 8·77-s − 4·79-s − 12·83-s − 4·85-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.20·11-s + 0.277·13-s + 0.970·17-s − 1.37·19-s + 1/5·25-s − 0.742·29-s + 1.07·31-s − 0.338·35-s + 0.328·37-s − 1.56·41-s − 1.21·43-s − 3/7·49-s − 0.549·53-s − 0.539·55-s + 0.520·59-s − 0.256·61-s − 0.124·65-s − 0.733·67-s + 0.949·71-s + 1.17·73-s + 0.911·77-s − 0.450·79-s − 1.31·83-s − 0.433·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.416320984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.416320984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.80714309705387, −14.46037629351189, −13.93707085453610, −13.30426709648742, −12.74164443615972, −12.09410957874151, −11.75739014647127, −11.25021674656278, −10.77521322980951, −10.03480237246845, −9.647516767834423, −8.820341854764429, −8.372580739283517, −8.054210072743076, −7.274839651367673, −6.666693028222297, −6.228952997749889, −5.483562051741089, −4.750446899477233, −4.308428951852800, −3.609054969754877, −3.103807097014469, −1.964917190790346, −1.515935578786788, −0.5913301424842526,
0.5913301424842526, 1.515935578786788, 1.964917190790346, 3.103807097014469, 3.609054969754877, 4.308428951852800, 4.750446899477233, 5.483562051741089, 6.228952997749889, 6.666693028222297, 7.274839651367673, 8.054210072743076, 8.372580739283517, 8.820341854764429, 9.647516767834423, 10.03480237246845, 10.77521322980951, 11.25021674656278, 11.75739014647127, 12.09410957874151, 12.74164443615972, 13.30426709648742, 13.93707085453610, 14.46037629351189, 14.80714309705387
