L(s) = 1 | + 2.28·3-s + 0.350·5-s + 2.23·9-s + 3.11·11-s − 3.97·13-s + 0.801·15-s − 2.70·17-s − 0.682·19-s − 4.11·23-s − 4.87·25-s − 1.74·27-s − 6.74·29-s + 0.465·31-s + 7.11·33-s − 4.33·37-s − 9.09·39-s + 41-s − 2.58·43-s + 0.783·45-s + 5.96·47-s − 6.18·51-s + 2.04·53-s + 1.08·55-s − 1.56·57-s − 10.3·59-s + 2.87·61-s − 1.39·65-s + ⋯ |
L(s) = 1 | + 1.32·3-s + 0.156·5-s + 0.746·9-s + 0.937·11-s − 1.10·13-s + 0.206·15-s − 0.655·17-s − 0.156·19-s − 0.858·23-s − 0.975·25-s − 0.335·27-s − 1.25·29-s + 0.0835·31-s + 1.23·33-s − 0.713·37-s − 1.45·39-s + 0.156·41-s − 0.394·43-s + 0.116·45-s + 0.870·47-s − 0.865·51-s + 0.280·53-s + 0.146·55-s − 0.206·57-s − 1.35·59-s + 0.367·61-s − 0.172·65-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 | \( 1 - 2.28T + 3T^{2} \) |
| 5 | \( 1 - 0.350T + 5T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 0.682T + 19T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 - 0.465T + 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 3.23T + 71T^{2} \) |
| 73 | \( 1 + 0.599T + 73T^{2} \) |
| 79 | \( 1 + 0.856T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 3.40T + 97T^{2} \) |
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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
−7.55542516834965651869144694309, −7.02236961513498994477782064917, −6.15590167335115749170414728471, −5.42048758459603692699263487718, −4.34597489349817341221998411757, −3.89631248080611500142325620608, −3.04901519186649027345077020051, −2.19686919485782070783503028282, −1.69361825329777957033844833901, 0,
1.69361825329777957033844833901, 2.19686919485782070783503028282, 3.04901519186649027345077020051, 3.89631248080611500142325620608, 4.34597489349817341221998411757, 5.42048758459603692699263487718, 6.15590167335115749170414728471, 7.02236961513498994477782064917, 7.55542516834965651869144694309