| L(s) = 1 | + 1.96·3-s + 3.24·5-s + 7-s + 0.855·9-s + 5.08·11-s − 0.544·13-s + 6.36·15-s + 0.841·17-s − 19-s + 1.96·21-s − 1.69·23-s + 5.51·25-s − 4.21·27-s + 10.0·29-s + 1.15·31-s + 9.98·33-s + 3.24·35-s − 4.21·37-s − 1.06·39-s − 3.95·41-s − 3.61·43-s + 2.77·45-s + 4.58·47-s + 49-s + 1.65·51-s − 5.97·53-s + 16.4·55-s + ⋯ |
| L(s) = 1 | + 1.13·3-s + 1.44·5-s + 0.377·7-s + 0.285·9-s + 1.53·11-s − 0.151·13-s + 1.64·15-s + 0.204·17-s − 0.229·19-s + 0.428·21-s − 0.352·23-s + 1.10·25-s − 0.810·27-s + 1.86·29-s + 0.208·31-s + 1.73·33-s + 0.547·35-s − 0.692·37-s − 0.171·39-s − 0.617·41-s − 0.551·43-s + 0.413·45-s + 0.669·47-s + 0.142·49-s + 0.231·51-s − 0.820·53-s + 2.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.417188686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.417188686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 + 0.544T + 13T^{2} \) |
| 17 | \( 1 - 0.841T + 17T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 1.15T + 31T^{2} \) |
| 37 | \( 1 + 4.21T + 37T^{2} \) |
| 41 | \( 1 + 3.95T + 41T^{2} \) |
| 43 | \( 1 + 3.61T + 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 + 8.97T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 - 9.68T + 71T^{2} \) |
| 73 | \( 1 - 6.33T + 73T^{2} \) |
| 79 | \( 1 + 4.79T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.525975900051228910695341706242, −7.889507666325330803539005824610, −6.76036384142486335521652152983, −6.35992102719949393071831264701, −5.47004334430834738318845906121, −4.58965232272853715510031551715, −3.66565017534332581416300496439, −2.79916173469995303855216335271, −1.98792739617034349664976151971, −1.28768996131314166089684227314,
1.28768996131314166089684227314, 1.98792739617034349664976151971, 2.79916173469995303855216335271, 3.66565017534332581416300496439, 4.58965232272853715510031551715, 5.47004334430834738318845906121, 6.35992102719949393071831264701, 6.76036384142486335521652152983, 7.889507666325330803539005824610, 8.525975900051228910695341706242
