| L(s) = 1 | + 3.10·3-s + 3.31·5-s + 6.61·9-s − 0.792·11-s − 6.10·13-s + 10.2·15-s − 0.423·17-s − 0.177·19-s + 5.55·23-s + 5.98·25-s + 11.2·27-s + 6.87·29-s − 1.03·31-s − 2.45·33-s + 10.9·37-s − 18.9·39-s + 41-s + 7.60·43-s + 21.9·45-s + 0.719·47-s − 1.31·51-s + 9.70·53-s − 2.62·55-s − 0.551·57-s − 8.31·59-s − 9.24·61-s − 20.2·65-s + ⋯ |
| L(s) = 1 | + 1.79·3-s + 1.48·5-s + 2.20·9-s − 0.238·11-s − 1.69·13-s + 2.65·15-s − 0.102·17-s − 0.0407·19-s + 1.15·23-s + 1.19·25-s + 2.15·27-s + 1.27·29-s − 0.186·31-s − 0.427·33-s + 1.79·37-s − 3.02·39-s + 0.156·41-s + 1.15·43-s + 3.26·45-s + 0.104·47-s − 0.183·51-s + 1.33·53-s − 0.353·55-s − 0.0730·57-s − 1.08·59-s − 1.18·61-s − 2.50·65-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
| \(L(1)\) |
\(\approx\) |
\(5.651884051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.651884051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 11 | \( 1 + 0.792T + 11T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 + 0.423T + 17T^{2} \) |
| 19 | \( 1 + 0.177T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 43 | \( 1 - 7.60T + 43T^{2} \) |
| 47 | \( 1 - 0.719T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 + 8.31T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 - 0.0105T + 79T^{2} \) |
| 83 | \( 1 + 7.04T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.68264132329346205591380096760, −7.44298084271734357594722984007, −6.57742581706559923862781493237, −5.78767249595236786918079672951, −4.81563782144914628341951428982, −4.37179628419310605345835103184, −3.04652438174028337177888135955, −2.64523055221906089544459157950, −2.13043063402296257741787024996, −1.14439937073538111489487272688,
1.14439937073538111489487272688, 2.13043063402296257741787024996, 2.64523055221906089544459157950, 3.04652438174028337177888135955, 4.37179628419310605345835103184, 4.81563782144914628341951428982, 5.78767249595236786918079672951, 6.57742581706559923862781493237, 7.44298084271734357594722984007, 7.68264132329346205591380096760
