L(s) = 1 | + 5-s + 4·7-s + 6·11-s − 4·13-s − 2·17-s − 23-s + 25-s − 8·29-s + 8·31-s + 4·35-s − 6·37-s + 2·41-s + 6·43-s + 12·47-s + 9·49-s − 6·53-s + 6·55-s − 4·59-s + 2·61-s − 4·65-s + 6·67-s + 6·71-s + 14·73-s + 24·77-s − 2·85-s − 12·89-s − 16·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.80·11-s − 1.10·13-s − 0.485·17-s − 0.208·23-s + 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.312·41-s + 0.914·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.809·55-s − 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.733·67-s + 0.712·71-s + 1.63·73-s + 2.73·77-s − 0.216·85-s − 1.27·89-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.075956968\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.075956968\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.73073234657050150623341480445, −7.17710859072532218036879145389, −6.44766158430502723771433890053, −5.71128825270245073853630312862, −4.92089415833609027730817093822, −4.37923231849502679376550358717, −3.66292423675929440929574123162, −2.36785677867917375107863105079, −1.81702010177320928428099208215, −0.911516956997658826024212794738,
0.911516956997658826024212794738, 1.81702010177320928428099208215, 2.36785677867917375107863105079, 3.66292423675929440929574123162, 4.37923231849502679376550358717, 4.92089415833609027730817093822, 5.71128825270245073853630312862, 6.44766158430502723771433890053, 7.17710859072532218036879145389, 7.73073234657050150623341480445