| L(s) = 1 | − 2-s + 4-s − 3.64·5-s − 8-s + 3.64·10-s − 3.64·11-s + 4.64·13-s + 16-s + 3.64·17-s − 2·19-s − 3.64·20-s + 3.64·22-s + 1.29·23-s + 8.29·25-s − 4.64·26-s − 2.35·29-s + 4.64·31-s − 32-s − 3.64·34-s − 11.9·37-s + 2·38-s + 3.64·40-s + 10.9·41-s + 5·43-s − 3.64·44-s − 1.29·46-s + 4.93·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.63·5-s − 0.353·8-s + 1.15·10-s − 1.09·11-s + 1.28·13-s + 0.250·16-s + 0.884·17-s − 0.458·19-s − 0.815·20-s + 0.777·22-s + 0.269·23-s + 1.65·25-s − 0.911·26-s − 0.437·29-s + 0.834·31-s − 0.176·32-s − 0.625·34-s − 1.96·37-s + 0.324·38-s + 0.576·40-s + 1.70·41-s + 0.762·43-s − 0.549·44-s − 0.190·46-s + 0.720·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 7.35T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 4.93T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.511816919468157650851283372132, −7.65284504606825201064579838134, −7.43037570817306926961562013433, −6.30812858848943826107343754921, −5.43549399869358257334137891093, −4.33602734939643909219002086839, −3.55058898791401079522136662345, −2.73722616980071727495934748614, −1.19058874761907471552965084871, 0,
1.19058874761907471552965084871, 2.73722616980071727495934748614, 3.55058898791401079522136662345, 4.33602734939643909219002086839, 5.43549399869358257334137891093, 6.30812858848943826107343754921, 7.43037570817306926961562013433, 7.65284504606825201064579838134, 8.511816919468157650851283372132
