L(s) = 1 | + 3-s − 5-s + 9-s − 6·13-s − 15-s − 6·17-s + 6·19-s + 23-s + 25-s + 27-s + 6·29-s + 8·31-s + 8·37-s − 6·39-s + 6·41-s − 4·43-s − 45-s + 4·47-s − 6·51-s + 2·53-s + 6·57-s − 12·59-s + 6·61-s + 6·65-s − 8·67-s + 69-s − 10·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.66·13-s − 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.31·37-s − 0.960·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s − 0.840·51-s + 0.274·53-s + 0.794·57-s − 1.56·59-s + 0.768·61-s + 0.744·65-s − 0.977·67-s + 0.120·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.526470015\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.526470015\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.43197030133644, −13.10146747267443, −12.37225658335295, −12.01578450914638, −11.69060072293402, −11.05239547935048, −10.49752323623249, −10.00372391862508, −9.477733720771555, −9.176490305029819, −8.558397630897365, −8.004474651002022, −7.508048891952259, −7.273850846605279, −6.483354097239727, −6.212195680661143, −5.172541421693466, −4.807995456519549, −4.409283524319724, −3.794715633478518, −2.875348620616266, −2.732302597740443, −2.111496065591041, −1.138733098059730, −0.4959158813366411,
0.4959158813366411, 1.138733098059730, 2.111496065591041, 2.732302597740443, 2.875348620616266, 3.794715633478518, 4.409283524319724, 4.807995456519549, 5.172541421693466, 6.212195680661143, 6.483354097239727, 7.273850846605279, 7.508048891952259, 8.004474651002022, 8.558397630897365, 9.176490305029819, 9.477733720771555, 10.00372391862508, 10.49752323623249, 11.05239547935048, 11.69060072293402, 12.01578450914638, 12.37225658335295, 13.10146747267443, 13.43197030133644