| L(s) = 1 | − 2.02·3-s + 3.89·5-s − 7-s + 1.10·9-s + 11-s + 13-s − 7.88·15-s + 7.08·17-s − 4.27·19-s + 2.02·21-s + 1.82·23-s + 10.1·25-s + 3.84·27-s − 1.97·29-s − 6.29·31-s − 2.02·33-s − 3.89·35-s + 1.05·37-s − 2.02·39-s − 5.53·41-s + 10.5·43-s + 4.29·45-s + 1.82·47-s + 49-s − 14.3·51-s − 0.942·53-s + 3.89·55-s + ⋯ |
| L(s) = 1 | − 1.16·3-s + 1.74·5-s − 0.377·7-s + 0.368·9-s + 0.301·11-s + 0.277·13-s − 2.03·15-s + 1.71·17-s − 0.981·19-s + 0.442·21-s + 0.380·23-s + 2.03·25-s + 0.739·27-s − 0.366·29-s − 1.13·31-s − 0.352·33-s − 0.657·35-s + 0.173·37-s − 0.324·39-s − 0.864·41-s + 1.60·43-s + 0.640·45-s + 0.265·47-s + 0.142·49-s − 2.00·51-s − 0.129·53-s + 0.524·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.759016806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759016806\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 17 | \( 1 - 7.08T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 + 6.29T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 + 5.53T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 + 0.942T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 6.79T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 3.31T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.683059099513923397351527583313, −7.50944228719318450506180072477, −6.62403167844065692755054566947, −6.13702434668375951411579769722, −5.53032194149231309287989098180, −5.15896786172717577116543078618, −3.91288019552618886025586561525, −2.83237768024002490110537049179, −1.79737790533281673615954118042, −0.837750622730866174328864606099,
0.837750622730866174328864606099, 1.79737790533281673615954118042, 2.83237768024002490110537049179, 3.91288019552618886025586561525, 5.15896786172717577116543078618, 5.53032194149231309287989098180, 6.13702434668375951411579769722, 6.62403167844065692755054566947, 7.50944228719318450506180072477, 8.683059099513923397351527583313
