| L(s) = 1 | − 1.04·2-s + 0.769·3-s − 0.918·4-s + 1.67·5-s − 0.800·6-s + 3.03·8-s − 2.40·9-s − 1.74·10-s − 0.537·11-s − 0.706·12-s + 1.28·15-s − 1.32·16-s + 5.62·17-s + 2.50·18-s − 2.01·19-s − 1.53·20-s + 0.558·22-s − 6.67·23-s + 2.33·24-s − 2.18·25-s − 4.16·27-s − 4.87·29-s − 1.34·30-s + 3.78·31-s − 4.69·32-s − 0.413·33-s − 5.85·34-s + ⋯ |
| L(s) = 1 | − 0.735·2-s + 0.444·3-s − 0.459·4-s + 0.749·5-s − 0.326·6-s + 1.07·8-s − 0.802·9-s − 0.551·10-s − 0.162·11-s − 0.203·12-s + 0.333·15-s − 0.330·16-s + 1.36·17-s + 0.590·18-s − 0.462·19-s − 0.344·20-s + 0.119·22-s − 1.39·23-s + 0.476·24-s − 0.437·25-s − 0.800·27-s − 0.905·29-s − 0.244·30-s + 0.680·31-s − 0.830·32-s − 0.0719·33-s − 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214559582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214559582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 3 | \( 1 - 0.769T + 3T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 11 | \( 1 + 0.537T + 11T^{2} \) |
| 17 | \( 1 - 5.62T + 17T^{2} \) |
| 19 | \( 1 + 2.01T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 + 4.87T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 + 1.38T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 - 3.79T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 8.07T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 + 9.59T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 8.26T + 79T^{2} \) |
| 83 | \( 1 + 9.42T + 83T^{2} \) |
| 89 | \( 1 - 2.13T + 89T^{2} \) |
| 97 | \( 1 - 6.44T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.966110810327896504126529368912, −7.46734787659404367151536854232, −6.41429295178192287516310023115, −5.61796062708592444025557482559, −5.27865525594764138986256878196, −4.10586187572090043273322440619, −3.52224001363292741103452601104, −2.40936230938446189966665092811, −1.76843324296167566049313273401, −0.59212576190429624918689564234,
0.59212576190429624918689564234, 1.76843324296167566049313273401, 2.40936230938446189966665092811, 3.52224001363292741103452601104, 4.10586187572090043273322440619, 5.27865525594764138986256878196, 5.61796062708592444025557482559, 6.41429295178192287516310023115, 7.46734787659404367151536854232, 7.966110810327896504126529368912
