| L(s) = 1 | + 3-s − 3.60i·5-s − 4.49i·7-s + 9-s + 0.890i·11-s + (−2.60 − 2.49i)13-s − 3.60i·15-s + 2·17-s + 4.49i·19-s − 4.49i·21-s − 1.78·23-s − 7.98·25-s + 27-s + 0.219·29-s + 2.71i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.61i·5-s − 1.69i·7-s + 0.333·9-s + 0.268i·11-s + (−0.722 − 0.691i)13-s − 0.930i·15-s + 0.485·17-s + 1.03i·19-s − 0.980i·21-s − 0.371·23-s − 1.59·25-s + 0.192·27-s + 0.0408·29-s + 0.487i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736180965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736180965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (2.60 + 2.49i)T \) |
| good | 5 | \( 1 + 3.60iT - 5T^{2} \) |
| 7 | \( 1 + 4.49iT - 7T^{2} \) |
| 11 | \( 1 - 0.890iT - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4.49iT - 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 - 0.219T + 29T^{2} \) |
| 31 | \( 1 - 2.71iT - 31T^{2} \) |
| 37 | \( 1 - 5.78iT - 37T^{2} \) |
| 41 | \( 1 + 10.8iT - 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.89iT - 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 8.09iT - 59T^{2} \) |
| 61 | \( 1 - 7.42T + 61T^{2} \) |
| 67 | \( 1 - 3.70iT - 67T^{2} \) |
| 71 | \( 1 + 5.87iT - 71T^{2} \) |
| 73 | \( 1 + 7.20iT - 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + 6.37iT - 89T^{2} \) |
| 97 | \( 1 + 15.2iT - 97T^{2} \) |
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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
−9.448361233268064717446974582593, −8.503614265402130867249403991798, −7.74955112244625516079948879644, −7.34131747606038750364717157329, −5.94789087454917716652316964970, −4.87475957929904451186531435475, −4.27123620144871678144696710412, −3.36809255112074406128377501545, −1.68662952918541800166150417226, −0.67934365049726557331540198012,
2.25639762846139152353864864929, 2.62804363828673781097324886435, 3.61134646750333181644177605412, 4.96022073342763300880977751613, 6.02567816036941881732805103686, 6.67619118973032381260416361798, 7.56352124484518773307925851025, 8.355246756903773589177758994083, 9.425585824441476253021108084067, 9.664575915526977139662894870434
