| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 11-s + 4·14-s + 16-s + 8·17-s + 19-s − 22-s + 3·23-s − 4·28-s + 29-s + 31-s − 32-s − 8·34-s − 2·37-s − 38-s + 10·41-s − 8·43-s + 44-s − 3·46-s + 9·49-s − 3·53-s + 4·56-s − 58-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 0.301·11-s + 1.06·14-s + 1/4·16-s + 1.94·17-s + 0.229·19-s − 0.213·22-s + 0.625·23-s − 0.755·28-s + 0.185·29-s + 0.179·31-s − 0.176·32-s − 1.37·34-s − 0.328·37-s − 0.162·38-s + 1.56·41-s − 1.21·43-s + 0.150·44-s − 0.442·46-s + 9/7·49-s − 0.412·53-s + 0.534·56-s − 0.131·58-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155007064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155007064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 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 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.69272616934500065227027719997, −7.20034947951988615899547228149, −6.46277252101632157710835502583, −5.91504680806510160620433365866, −5.20737114815027915943952969185, −4.04171245930655146527869249224, −3.20348658210524208005708423467, −2.85526100540655317757443528329, −1.50987751938155163694495652949, −0.61036834941842485857862232181,
0.61036834941842485857862232181, 1.50987751938155163694495652949, 2.85526100540655317757443528329, 3.20348658210524208005708423467, 4.04171245930655146527869249224, 5.20737114815027915943952969185, 5.91504680806510160620433365866, 6.46277252101632157710835502583, 7.20034947951988615899547228149, 7.69272616934500065227027719997
