L(s) = 1 | + 1.61·3-s − 0.618·5-s − 7-s − 0.381·9-s + 11-s − 13-s − 1.00·15-s + 0.854·17-s + 7.09·19-s − 1.61·21-s + 2.47·23-s − 4.61·25-s − 5.47·27-s + 1.61·33-s + 0.618·35-s + 2.76·37-s − 1.61·39-s − 11.7·41-s + 4.09·43-s + 0.236·45-s − 4·47-s + 49-s + 1.38·51-s + 7.61·53-s − 0.618·55-s + 11.4·57-s + 6·59-s + ⋯ |
L(s) = 1 | + 0.934·3-s − 0.276·5-s − 0.377·7-s − 0.127·9-s + 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.207·17-s + 1.62·19-s − 0.353·21-s + 0.515·23-s − 0.923·25-s − 1.05·27-s + 0.281·33-s + 0.104·35-s + 0.454·37-s − 0.259·39-s − 1.82·41-s + 0.623·43-s + 0.0351·45-s − 0.583·47-s + 0.142·49-s + 0.193·51-s + 1.04·53-s − 0.0833·55-s + 1.51·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.449061953\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.449061953\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 17 | \( 1 - 0.854T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.09T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 - 4.61T + 79T^{2} \) |
| 83 | \( 1 - 6.14T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.966619582513237075922645392857, −7.21519805821616025312639012810, −6.66528475089868918943734070071, −5.60999900109687516492151414278, −5.15151557811363224353330832781, −3.97206369200633276632331992072, −3.47333591434763346922513216324, −2.79781088439245053555135025834, −1.94136736697565553688181183716, −0.73081687646427890641279274082,
0.73081687646427890641279274082, 1.94136736697565553688181183716, 2.79781088439245053555135025834, 3.47333591434763346922513216324, 3.97206369200633276632331992072, 5.15151557811363224353330832781, 5.60999900109687516492151414278, 6.66528475089868918943734070071, 7.21519805821616025312639012810, 7.966619582513237075922645392857