| L(s) = 1 | − 2.47·3-s + 3.49·5-s + 3.13·9-s − 0.886·11-s + 6.85·13-s − 8.66·15-s − 3.02·17-s − 6.07·19-s − 4.52·23-s + 7.23·25-s − 0.340·27-s − 7.22·29-s − 9.72·31-s + 2.19·33-s + 7.05·37-s − 16.9·39-s + 41-s + 4.25·43-s + 10.9·45-s + 11.7·47-s + 7.50·51-s + 6.23·53-s − 3.10·55-s + 15.0·57-s − 0.453·59-s + 1.80·61-s + 23.9·65-s + ⋯ |
| L(s) = 1 | − 1.43·3-s + 1.56·5-s + 1.04·9-s − 0.267·11-s + 1.90·13-s − 2.23·15-s − 0.734·17-s − 1.39·19-s − 0.943·23-s + 1.44·25-s − 0.0654·27-s − 1.34·29-s − 1.74·31-s + 0.382·33-s + 1.16·37-s − 2.71·39-s + 0.156·41-s + 0.649·43-s + 1.63·45-s + 1.71·47-s + 1.05·51-s + 0.856·53-s − 0.418·55-s + 1.99·57-s − 0.0590·59-s + 0.231·61-s + 2.97·65-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
| \(L(1)\) |
\(\approx\) |
\(1.500368722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500368722\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 3.49T + 5T^{2} \) |
| 11 | \( 1 + 0.886T + 11T^{2} \) |
| 13 | \( 1 - 6.85T + 13T^{2} \) |
| 17 | \( 1 + 3.02T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 + 7.22T + 29T^{2} \) |
| 31 | \( 1 + 9.72T + 31T^{2} \) |
| 37 | \( 1 - 7.05T + 37T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 + 0.453T + 59T^{2} \) |
| 61 | \( 1 - 1.80T + 61T^{2} \) |
| 67 | \( 1 - 0.386T + 67T^{2} \) |
| 71 | \( 1 - 0.406T + 71T^{2} \) |
| 73 | \( 1 + 2.47T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 7.79T + 83T^{2} \) |
| 89 | \( 1 - 5.14T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.68876528525443375840765006860, −6.77746708269015948902814280258, −6.10936073678084597476666280722, −5.89945298106627894336002853753, −5.42303686816810579234418219989, −4.39094625091119506430416135954, −3.74805970462090102961887076095, −2.29810479627820692994666369160, −1.75017657156621081645102433399, −0.65713141235944615719411930414,
0.65713141235944615719411930414, 1.75017657156621081645102433399, 2.29810479627820692994666369160, 3.74805970462090102961887076095, 4.39094625091119506430416135954, 5.42303686816810579234418219989, 5.89945298106627894336002853753, 6.10936073678084597476666280722, 6.77746708269015948902814280258, 7.68876528525443375840765006860
