| L(s) = 1 | + 5.49·2-s + 22.1·4-s + 8.06·5-s + 32.6·7-s + 77.6·8-s + 44.2·10-s − 21.5·11-s − 50.2·13-s + 179.·14-s + 249.·16-s − 11.5·17-s + 13.6·19-s + 178.·20-s − 118.·22-s + 150.·23-s − 59.9·25-s − 275.·26-s + 722.·28-s − 6.42·29-s − 123.·31-s + 746.·32-s − 63.5·34-s + 263.·35-s − 81.6·37-s + 74.9·38-s + 626.·40-s + 402.·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.76·4-s + 0.721·5-s + 1.76·7-s + 3.43·8-s + 1.40·10-s − 0.591·11-s − 1.07·13-s + 3.41·14-s + 3.89·16-s − 0.165·17-s + 0.164·19-s + 1.99·20-s − 1.14·22-s + 1.36·23-s − 0.479·25-s − 2.08·26-s + 4.87·28-s − 0.0411·29-s − 0.716·31-s + 4.12·32-s − 0.320·34-s + 1.27·35-s − 0.362·37-s + 0.320·38-s + 2.47·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.36645200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.36645200\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 5.49T + 8T^{2} \) |
| 5 | \( 1 - 8.06T + 125T^{2} \) |
| 7 | \( 1 - 32.6T + 343T^{2} \) |
| 11 | \( 1 + 21.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6.42T + 2.43e4T^{2} \) |
| 31 | \( 1 + 123.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 81.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 402.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 189.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 166.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 264.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 283.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.07e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 386.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 329.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 807.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.52e3T + 9.12e5T^{2} \) |
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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
−8.477404107187989935168973647930, −7.40820208649311481275404391522, −7.22162898514858142562819961197, −5.88145285301032526900304367644, −5.37423024109358571317908813988, −4.80861034675229238761166104795, −4.14834695473497160899801980553, −2.82860699967217948051044160723, −2.19436497869985404313380196317, −1.37285985207896187281875250537,
1.37285985207896187281875250537, 2.19436497869985404313380196317, 2.82860699967217948051044160723, 4.14834695473497160899801980553, 4.80861034675229238761166104795, 5.37423024109358571317908813988, 5.88145285301032526900304367644, 7.22162898514858142562819961197, 7.40820208649311481275404391522, 8.477404107187989935168973647930
