| L(s) = 1 | + 2.81·3-s − 5-s − 1.03·7-s + 4.90·9-s + 3.44·13-s − 2.81·15-s + 2.39·17-s + 7.66·19-s − 2.89·21-s − 2.45·23-s + 25-s + 5.35·27-s − 5.95·29-s + 3.68·31-s + 1.03·35-s + 5.95·37-s + 9.69·39-s + 3.93·41-s + 7.64·43-s − 4.90·45-s − 5.84·47-s − 5.93·49-s + 6.74·51-s − 11.8·53-s + 21.5·57-s − 2.94·59-s + 2.48·61-s + ⋯ |
| L(s) = 1 | + 1.62·3-s − 0.447·5-s − 0.389·7-s + 1.63·9-s + 0.956·13-s − 0.725·15-s + 0.581·17-s + 1.75·19-s − 0.632·21-s − 0.512·23-s + 0.200·25-s + 1.03·27-s − 1.10·29-s + 0.662·31-s + 0.174·35-s + 0.979·37-s + 1.55·39-s + 0.614·41-s + 1.16·43-s − 0.731·45-s − 0.852·47-s − 0.848·49-s + 0.944·51-s − 1.62·53-s + 2.85·57-s − 0.383·59-s + 0.317·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.005334318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.005334318\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.81T + 3T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 13 | \( 1 - 3.44T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 + 5.95T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 43 | \( 1 - 7.64T + 43T^{2} \) |
| 47 | \( 1 + 5.84T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 - 2.48T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 + 2.02T + 71T^{2} \) |
| 73 | \( 1 + 0.825T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 - 2.44T + 97T^{2} \) |
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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
−7.82065498999951856059440757235, −7.35052549828193800575956199436, −6.41024286333495426779443411643, −5.70151878308645160128692960256, −4.70285813562278213383167687716, −3.85621773570162230605040855742, −3.35743077262298665288595554634, −2.86840009337093447738587200508, −1.83370724456034993769278985754, −0.917235500499891970474969098608,
0.917235500499891970474969098608, 1.83370724456034993769278985754, 2.86840009337093447738587200508, 3.35743077262298665288595554634, 3.85621773570162230605040855742, 4.70285813562278213383167687716, 5.70151878308645160128692960256, 6.41024286333495426779443411643, 7.35052549828193800575956199436, 7.82065498999951856059440757235
