| L(s) = 1 | − 3-s − 1.38·5-s − 0.260·7-s + 9-s − 3.57·11-s − 1.29·13-s + 1.38·15-s − 4.24·17-s + 5.37·19-s + 0.260·21-s + 5.60·23-s − 3.09·25-s − 27-s + 2.23·29-s − 9.62·31-s + 3.57·33-s + 0.359·35-s + 0.0808·37-s + 1.29·39-s + 4.29·41-s − 10.5·43-s − 1.38·45-s − 6.20·47-s − 6.93·49-s + 4.24·51-s + 12.3·53-s + 4.94·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.617·5-s − 0.0983·7-s + 0.333·9-s − 1.07·11-s − 0.360·13-s + 0.356·15-s − 1.02·17-s + 1.23·19-s + 0.0567·21-s + 1.16·23-s − 0.618·25-s − 0.192·27-s + 0.414·29-s − 1.72·31-s + 0.623·33-s + 0.0607·35-s + 0.0132·37-s + 0.208·39-s + 0.670·41-s − 1.60·43-s − 0.205·45-s − 0.904·47-s − 0.990·49-s + 0.594·51-s + 1.70·53-s + 0.666·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8077449353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8077449353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 0.260T + 7T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 5.60T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 + 9.62T + 31T^{2} \) |
| 37 | \( 1 - 0.0808T + 37T^{2} \) |
| 41 | \( 1 - 4.29T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 3.18T + 59T^{2} \) |
| 61 | \( 1 + 6.40T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 - 8.61T + 89T^{2} \) |
| 97 | \( 1 - 0.0718T + 97T^{2} \) |
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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
−8.359231971493027705803495641258, −7.55733697983232646022855715076, −7.12136451343071205209141620796, −6.26804722232459302639189203439, −5.20459775805842277548728898485, −4.97311952553631150847699623503, −3.84615926117287062146099665392, −3.03865102709160694226536866905, −1.95113895340553256828869741685, −0.51230278561142327902849687728,
0.51230278561142327902849687728, 1.95113895340553256828869741685, 3.03865102709160694226536866905, 3.84615926117287062146099665392, 4.97311952553631150847699623503, 5.20459775805842277548728898485, 6.26804722232459302639189203439, 7.12136451343071205209141620796, 7.55733697983232646022855715076, 8.359231971493027705803495641258
