| L(s) = 1 | − 3-s + 0.950·5-s − 7-s + 9-s − 1.41·11-s + 1.22·13-s − 0.950·15-s + 6.23·17-s − 19-s + 21-s + 2.69·23-s − 4.09·25-s − 27-s − 5.55·29-s − 10.2·31-s + 1.41·33-s − 0.950·35-s − 5.74·37-s − 1.22·39-s + 11.3·41-s − 5.37·43-s + 0.950·45-s − 2.09·47-s + 49-s − 6.23·51-s − 5.55·53-s − 1.34·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.425·5-s − 0.377·7-s + 0.333·9-s − 0.426·11-s + 0.340·13-s − 0.245·15-s + 1.51·17-s − 0.229·19-s + 0.218·21-s + 0.562·23-s − 0.819·25-s − 0.192·27-s − 1.03·29-s − 1.84·31-s + 0.246·33-s − 0.160·35-s − 0.943·37-s − 0.196·39-s + 1.78·41-s − 0.819·43-s + 0.141·45-s − 0.304·47-s + 0.142·49-s − 0.872·51-s − 0.762·53-s − 0.181·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.950T + 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 + 5.55T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 + 5.55T + 53T^{2} \) |
| 59 | \( 1 - 9.20T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 9.88T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 - 0.0982T + 73T^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 + 18.1T + 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−7.52435625222666788793650211741, −7.03230892395569087537664788936, −6.04674120845036919494440240931, −5.61434989593355255464982787673, −5.06455000304411680079372921831, −3.89921785389450318262256104075, −3.34147496503317129503768408421, −2.19630995776106282588695484076, −1.27618542955064205748368443249, 0,
1.27618542955064205748368443249, 2.19630995776106282588695484076, 3.34147496503317129503768408421, 3.89921785389450318262256104075, 5.06455000304411680079372921831, 5.61434989593355255464982787673, 6.04674120845036919494440240931, 7.03230892395569087537664788936, 7.52435625222666788793650211741
