| L(s) = 1 | + 1.01·2-s + 2.28·3-s − 0.971·4-s + 5-s + 2.32·6-s − 7-s − 3.01·8-s + 2.23·9-s + 1.01·10-s + 0.810·11-s − 2.22·12-s − 2.07·13-s − 1.01·14-s + 2.28·15-s − 1.11·16-s + 1.83·17-s + 2.26·18-s − 0.366·19-s − 0.971·20-s − 2.28·21-s + 0.822·22-s − 4.19·23-s − 6.89·24-s + 25-s − 2.10·26-s − 1.75·27-s + 0.971·28-s + ⋯ |
| L(s) = 1 | + 0.717·2-s + 1.32·3-s − 0.485·4-s + 0.447·5-s + 0.947·6-s − 0.377·7-s − 1.06·8-s + 0.744·9-s + 0.320·10-s + 0.244·11-s − 0.641·12-s − 0.574·13-s − 0.271·14-s + 0.590·15-s − 0.278·16-s + 0.443·17-s + 0.533·18-s − 0.0841·19-s − 0.217·20-s − 0.499·21-s + 0.175·22-s − 0.873·23-s − 1.40·24-s + 0.200·25-s − 0.411·26-s − 0.337·27-s + 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 1.01T + 2T^{2} \) |
| 3 | \( 1 - 2.28T + 3T^{2} \) |
| 11 | \( 1 - 0.810T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 + 0.366T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 + 9.78T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 + 9.53T + 43T^{2} \) |
| 47 | \( 1 - 5.48T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 - 4.68T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 0.617T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 - 5.26T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 1.96T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
| 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
−7.55389474439063803444692776200, −6.78070410282967918311133478231, −5.96852210691419992456643494167, −5.33092246284940936627510858231, −4.53246523459737007797869392693, −3.73594379039672201786816218533, −3.24115640759345522789766228146, −2.50881131810792894730698702926, −1.62220466650759511273955303888, 0,
1.62220466650759511273955303888, 2.50881131810792894730698702926, 3.24115640759345522789766228146, 3.73594379039672201786816218533, 4.53246523459737007797869392693, 5.33092246284940936627510858231, 5.96852210691419992456643494167, 6.78070410282967918311133478231, 7.55389474439063803444692776200
