| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s − 4·10-s + 5·11-s − 13-s − 16-s − 3·17-s − 5·19-s + 4·20-s + 5·22-s − 6·23-s + 11·25-s − 26-s − 7·29-s + 5·32-s − 3·34-s − 5·38-s + 12·40-s − 8·41-s + 2·43-s − 5·44-s − 6·46-s − 9·47-s + 11·50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s − 1.26·10-s + 1.50·11-s − 0.277·13-s − 1/4·16-s − 0.727·17-s − 1.14·19-s + 0.894·20-s + 1.06·22-s − 1.25·23-s + 11/5·25-s − 0.196·26-s − 1.29·29-s + 0.883·32-s − 0.514·34-s − 0.811·38-s + 1.89·40-s − 1.24·41-s + 0.304·43-s − 0.753·44-s − 0.884·46-s − 1.31·47-s + 1.55·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7432768101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7432768101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.268614211863629364088154679465, −7.35799786994427946433195235912, −6.63262762075145158653477046621, −6.03327867692983815181241185107, −4.85836809805781131443132078574, −4.35786337680704206921048143314, −3.78287528087977959203940349759, −3.35920429206950222743094784345, −1.94972193301931438855961779821, −0.39983215375450315530778404089,
0.39983215375450315530778404089, 1.94972193301931438855961779821, 3.35920429206950222743094784345, 3.78287528087977959203940349759, 4.35786337680704206921048143314, 4.85836809805781131443132078574, 6.03327867692983815181241185107, 6.63262762075145158653477046621, 7.35799786994427946433195235912, 8.268614211863629364088154679465
