| L(s) = 1 | + 2·3-s − 5-s + 7-s + 9-s + 6·11-s − 13-s − 2·15-s − 2·17-s + 2·19-s + 2·21-s − 2·23-s + 25-s − 4·27-s + 10·29-s − 6·31-s + 12·33-s − 35-s + 6·37-s − 2·39-s − 2·41-s + 10·43-s − 45-s + 49-s − 4·51-s + 14·53-s − 6·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s − 0.516·15-s − 0.485·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s − 1.07·31-s + 2.08·33-s − 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.312·41-s + 1.52·43-s − 0.149·45-s + 1/7·49-s − 0.560·51-s + 1.92·53-s − 0.809·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.049171595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.049171595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.609838383762858048247573806734, −7.925245387849671171025169939212, −7.19187209492735217735177440332, −6.50741785377801621029840038062, −5.54042150575646489689069819008, −4.32964042901436714897500918195, −3.95758346607943726243247521797, −3.00339694126723401472697619039, −2.13379787825443665400985068360, −1.02052801348837377914168880582,
1.02052801348837377914168880582, 2.13379787825443665400985068360, 3.00339694126723401472697619039, 3.95758346607943726243247521797, 4.32964042901436714897500918195, 5.54042150575646489689069819008, 6.50741785377801621029840038062, 7.19187209492735217735177440332, 7.925245387849671171025169939212, 8.609838383762858048247573806734
