| L(s) = 1 | − 3-s + 2.24·5-s − 1.35·7-s + 9-s − 0.445·11-s − 2.24·15-s − 1.80·17-s − 2.69·19-s + 1.35·21-s − 2.24·23-s + 0.0489·25-s − 27-s + 4.65·29-s + 8.25·31-s + 0.445·33-s − 3.04·35-s + 1.41·37-s − 9.03·41-s − 9.09·43-s + 2.24·45-s − 7.58·47-s − 5.15·49-s + 1.80·51-s + 11.4·53-s − 55-s + 2.69·57-s − 6.93·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.00·5-s − 0.512·7-s + 0.333·9-s − 0.134·11-s − 0.580·15-s − 0.437·17-s − 0.617·19-s + 0.296·21-s − 0.468·23-s + 0.00978·25-s − 0.192·27-s + 0.864·29-s + 1.48·31-s + 0.0774·33-s − 0.515·35-s + 0.233·37-s − 1.41·41-s − 1.38·43-s + 0.334·45-s − 1.10·47-s − 0.736·49-s + 0.252·51-s + 1.57·53-s − 0.134·55-s + 0.356·57-s − 0.902·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.24T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 0.445T + 11T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 - 8.25T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 + 9.03T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 6.93T + 59T^{2} \) |
| 61 | \( 1 + 0.868T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 3.35T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 3.19T + 83T^{2} \) |
| 89 | \( 1 + 0.454T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.201881502379814156046111754501, −7.10024022800684862487641383880, −6.34589796875202832984321461835, −6.10822156003934683279316172640, −5.08485776822898395825179018634, −4.48071430731100167982213566931, −3.33816472512706676692930349556, −2.37668970048841680958355516346, −1.44598569287685472879061441316, 0,
1.44598569287685472879061441316, 2.37668970048841680958355516346, 3.33816472512706676692930349556, 4.48071430731100167982213566931, 5.08485776822898395825179018634, 6.10822156003934683279316172640, 6.34589796875202832984321461835, 7.10024022800684862487641383880, 8.201881502379814156046111754501
