| L(s) = 1 | − 2·3-s − 2·4-s + 9-s + 3·11-s + 4·12-s − 4·13-s + 4·16-s − 3·17-s − 19-s + 4·27-s + 6·29-s + 4·31-s − 6·33-s − 2·36-s − 2·37-s + 8·39-s + 6·41-s + 43-s − 6·44-s − 3·47-s − 8·48-s + 6·51-s + 8·52-s − 12·53-s + 2·57-s + 6·59-s + 61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s + 16-s − 0.727·17-s − 0.229·19-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 1.04·33-s − 1/3·36-s − 0.328·37-s + 1.28·39-s + 0.937·41-s + 0.152·43-s − 0.904·44-s − 0.437·47-s − 1.15·48-s + 0.840·51-s + 1.10·52-s − 1.64·53-s + 0.264·57-s + 0.781·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6976059389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6976059389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.60105149980105, −14.68767825704058, −14.44412559456336, −13.90096636339785, −13.22805270775277, −12.63126029510127, −12.14358432954737, −11.83602145314449, −11.05134607347158, −10.65088717756469, −9.820209961977886, −9.602568605160957, −8.796233272763829, −8.356700911491885, −7.600780893028440, −6.794184153090423, −6.359778585514856, −5.765118963344202, −4.908302191378777, −4.735138411618941, −4.062517745702848, −3.186133804008821, −2.285629290500670, −1.155768627463700, −0.4220670573354645,
0.4220670573354645, 1.155768627463700, 2.285629290500670, 3.186133804008821, 4.062517745702848, 4.735138411618941, 4.908302191378777, 5.765118963344202, 6.359778585514856, 6.794184153090423, 7.600780893028440, 8.356700911491885, 8.796233272763829, 9.602568605160957, 9.820209961977886, 10.65088717756469, 11.05134607347158, 11.83602145314449, 12.14358432954737, 12.63126029510127, 13.22805270775277, 13.90096636339785, 14.44412559456336, 14.68767825704058, 15.60105149980105
