| L(s) = 1 | − 19·7-s − 22·11-s + 13-s + 58·17-s − 53·19-s − 58·23-s − 22·29-s − 35·31-s − 270·37-s + 468·41-s − 431·43-s + 230·47-s + 18·49-s − 446·59-s + 127·61-s − 811·67-s − 36·71-s + 522·73-s + 418·77-s + 1.36e3·79-s + 1.13e3·83-s − 144·89-s − 19·91-s − 1.07e3·97-s + 1.44e3·101-s + 124·103-s + 432·107-s + ⋯ |
| L(s) = 1 | − 1.02·7-s − 0.603·11-s + 0.0213·13-s + 0.827·17-s − 0.639·19-s − 0.525·23-s − 0.140·29-s − 0.202·31-s − 1.19·37-s + 1.78·41-s − 1.52·43-s + 0.713·47-s + 0.0524·49-s − 0.984·59-s + 0.266·61-s − 1.47·67-s − 0.0601·71-s + 0.836·73-s + 0.618·77-s + 1.94·79-s + 1.50·83-s − 0.171·89-s − 0.0218·91-s − 1.12·97-s + 1.41·101-s + 0.118·103-s + 0.390·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.238106820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238106820\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 - T + p^{3} T^{2} \) |
| 17 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 53 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 22 T + p^{3} T^{2} \) |
| 31 | \( 1 + 35 T + p^{3} T^{2} \) |
| 37 | \( 1 + 270 T + p^{3} T^{2} \) |
| 41 | \( 1 - 468 T + p^{3} T^{2} \) |
| 43 | \( 1 + 431 T + p^{3} T^{2} \) |
| 47 | \( 1 - 230 T + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + 446 T + p^{3} T^{2} \) |
| 61 | \( 1 - 127 T + p^{3} T^{2} \) |
| 67 | \( 1 + 811 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 - 522 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1368 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1138 T + p^{3} T^{2} \) |
| 89 | \( 1 + 144 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1079 T + p^{3} T^{2} \) |
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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
−8.992903842154091410157278655306, −8.081212738628160374206304051403, −7.36824544654995206314962087614, −6.45062428442972188329232864238, −5.80127486354274174766813245041, −4.86957640204838498499970468407, −3.76529621737892386171194266612, −3.02609427807122887681750241964, −1.94545679460177814690827762679, −0.50411310662759094186357961075,
0.50411310662759094186357961075, 1.94545679460177814690827762679, 3.02609427807122887681750241964, 3.76529621737892386171194266612, 4.86957640204838498499970468407, 5.80127486354274174766813245041, 6.45062428442972188329232864238, 7.36824544654995206314962087614, 8.081212738628160374206304051403, 8.992903842154091410157278655306
