L(s) = 1 | + (−0.0747 − 0.997i)3-s + (0.365 + 0.930i)7-s + (−0.988 + 0.149i)9-s + (1.55 + 1.24i)13-s + (−0.365 + 0.632i)19-s + (0.900 − 0.433i)21-s + (−0.365 + 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.733 − 1.26i)31-s + (1.40 + 1.29i)37-s + (1.12 − 1.64i)39-s + (0.129 + 0.268i)43-s + (−0.733 + 0.680i)49-s + (0.658 + 0.317i)57-s + (1.32 − 1.42i)61-s + ⋯ |
L(s) = 1 | + (−0.0747 − 0.997i)3-s + (0.365 + 0.930i)7-s + (−0.988 + 0.149i)9-s + (1.55 + 1.24i)13-s + (−0.365 + 0.632i)19-s + (0.900 − 0.433i)21-s + (−0.365 + 0.930i)25-s + (0.222 + 0.974i)27-s + (−0.733 − 1.26i)31-s + (1.40 + 1.29i)37-s + (1.12 − 1.64i)39-s + (0.129 + 0.268i)43-s + (−0.733 + 0.680i)49-s + (0.658 + 0.317i)57-s + (1.32 − 1.42i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.201519300\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.201519300\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0747 + 0.997i)T \) |
| 7 | \( 1 + (-0.365 - 0.930i)T \) |
good | 5 | \( 1 + (0.365 - 0.930i)T^{2} \) |
| 11 | \( 1 + (0.955 + 0.294i)T^{2} \) |
| 13 | \( 1 + (-1.55 - 1.24i)T + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 19 | \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.826 + 0.563i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2} \) |
| 41 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.129 - 0.268i)T + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 53 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 61 | \( 1 + (-1.32 + 1.42i)T + (-0.0747 - 0.997i)T^{2} \) |
| 67 | \( 1 + (-0.975 + 0.563i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.548 + 0.215i)T + (0.733 + 0.680i)T^{2} \) |
| 79 | \( 1 + (1.17 + 0.680i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.955 - 0.294i)T^{2} \) |
| 97 | \( 1 + 1.56iT - T^{2} \) |
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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
−9.008993645208549521377150953934, −8.336135658810567062941441277639, −7.78538640250188705288092586806, −6.73556753863170882185649213836, −6.10224929689768634199589096837, −5.57002130803790264701343141676, −4.36755157267045806992971535226, −3.33585577259520151545764916482, −2.13410371155600160100396561580, −1.45517705064671732413871003382,
0.922406693256429764080668415927, 2.62550695556913035409182859164, 3.70931437018607797475832413791, 4.15165539444424034719075942233, 5.20033771616219475479925496357, 5.87245781321524210460075170343, 6.77638772647952413920295204925, 7.82602785107495850989079437457, 8.460638004736058605206402887227, 9.102652573111531332596180037475