| L(s) = 1 | − 2.75·3-s − 2.20·5-s + 1.49·7-s + 4.58·9-s − 1.70·13-s + 6.08·15-s + 1.96·17-s + 19-s − 4.11·21-s − 1.66·23-s − 0.123·25-s − 4.36·27-s + 0.488·29-s − 2.05·31-s − 3.29·35-s + 0.415·37-s + 4.68·39-s − 5.96·41-s + 4.80·43-s − 10.1·45-s − 5.74·47-s − 4.76·49-s − 5.40·51-s + 6.74·53-s − 2.75·57-s + 5.44·59-s + 9.18·61-s + ⋯ |
| L(s) = 1 | − 1.59·3-s − 0.987·5-s + 0.564·7-s + 1.52·9-s − 0.472·13-s + 1.57·15-s + 0.475·17-s + 0.229·19-s − 0.897·21-s − 0.346·23-s − 0.0247·25-s − 0.839·27-s + 0.0907·29-s − 0.368·31-s − 0.557·35-s + 0.0682·37-s + 0.750·39-s − 0.931·41-s + 0.733·43-s − 1.50·45-s − 0.837·47-s − 0.681·49-s − 0.756·51-s + 0.925·53-s − 0.364·57-s + 0.708·59-s + 1.17·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 1.96T + 17T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 - 0.488T + 29T^{2} \) |
| 31 | \( 1 + 2.05T + 31T^{2} \) |
| 37 | \( 1 - 0.415T + 37T^{2} \) |
| 41 | \( 1 + 5.96T + 41T^{2} \) |
| 43 | \( 1 - 4.80T + 43T^{2} \) |
| 47 | \( 1 + 5.74T + 47T^{2} \) |
| 53 | \( 1 - 6.74T + 53T^{2} \) |
| 59 | \( 1 - 5.44T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 + 0.0250T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 6.05T + 79T^{2} \) |
| 83 | \( 1 + 3.29T + 83T^{2} \) |
| 89 | \( 1 - 8.67T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.24780224801924379715121998517, −6.76341819479558779417555394394, −5.85023330471759100390148290452, −5.35515347316483944058194257009, −4.67711532791518657517074482041, −4.10256153626101839473596284576, −3.23611690779023891009193138221, −1.94757076806438493546655970109, −0.903142370041745746839632615568, 0,
0.903142370041745746839632615568, 1.94757076806438493546655970109, 3.23611690779023891009193138221, 4.10256153626101839473596284576, 4.67711532791518657517074482041, 5.35515347316483944058194257009, 5.85023330471759100390148290452, 6.76341819479558779417555394394, 7.24780224801924379715121998517
