| L(s) = 1 | + 2.10·3-s + 1.26·5-s + 1.42·9-s − 3.91·11-s − 2.38·13-s + 2.65·15-s + 6.14·17-s − 5.49·19-s − 0.104·23-s − 3.40·25-s − 3.30·27-s − 2.65·29-s − 3.26·31-s − 8.24·33-s + 0.993·37-s − 5.02·39-s − 41-s + 8.14·43-s + 1.80·45-s − 1.90·47-s + 12.9·51-s + 1.26·53-s − 4.94·55-s − 11.5·57-s − 8.24·59-s − 6.89·61-s − 3.01·65-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 0.564·5-s + 0.475·9-s − 1.18·11-s − 0.662·13-s + 0.685·15-s + 1.49·17-s − 1.25·19-s − 0.0217·23-s − 0.681·25-s − 0.636·27-s − 0.493·29-s − 0.585·31-s − 1.43·33-s + 0.163·37-s − 0.804·39-s − 0.156·41-s + 1.24·43-s + 0.268·45-s − 0.277·47-s + 1.81·51-s + 0.173·53-s − 0.666·55-s − 1.53·57-s − 1.07·59-s − 0.882·61-s − 0.373·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 - 1.26T + 5T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 23 | \( 1 + 0.104T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 3.26T + 31T^{2} \) |
| 37 | \( 1 - 0.993T + 37T^{2} \) |
| 43 | \( 1 - 8.14T + 43T^{2} \) |
| 47 | \( 1 + 1.90T + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 - 8.69T + 67T^{2} \) |
| 71 | \( 1 - 0.827T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 8.31T + 97T^{2} \) |
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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
−7.73002016807971792781715768942, −7.04508624431919277339873670829, −5.87755293901933778278015736775, −5.56633804323706040792670487935, −4.58181666740488020750320269215, −3.74003066312526913286915118054, −2.91600660428906794924350844145, −2.37736438033560084425350544799, −1.61690575657937804790536010539, 0,
1.61690575657937804790536010539, 2.37736438033560084425350544799, 2.91600660428906794924350844145, 3.74003066312526913286915118054, 4.58181666740488020750320269215, 5.56633804323706040792670487935, 5.87755293901933778278015736775, 7.04508624431919277339873670829, 7.73002016807971792781715768942
