| L(s) = 1 | − 5i·3-s + 9·5-s − 16·9-s − 3i·11-s + 16·13-s − 45i·15-s − 7·17-s + 11i·19-s − 19i·23-s + 56·25-s + 35i·27-s − 32·29-s − 11i·31-s − 15·33-s − 37-s + ⋯ |
| L(s) = 1 | − 1.66i·3-s + 1.80·5-s − 1.77·9-s − 0.272i·11-s + 1.23·13-s − 3i·15-s − 0.411·17-s + 0.578i·19-s − 0.826i·23-s + 2.24·25-s + 1.29i·27-s − 1.10·29-s − 0.354i·31-s − 0.454·33-s − 0.0270·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.766895739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766895739\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5iT - 9T^{2} \) |
| 5 | \( 1 - 9T + 25T^{2} \) |
| 11 | \( 1 + 3iT - 121T^{2} \) |
| 13 | \( 1 - 16T + 169T^{2} \) |
| 17 | \( 1 + 7T + 289T^{2} \) |
| 19 | \( 1 - 11iT - 361T^{2} \) |
| 23 | \( 1 + 19iT - 529T^{2} \) |
| 29 | \( 1 + 32T + 841T^{2} \) |
| 31 | \( 1 + 11iT - 961T^{2} \) |
| 37 | \( 1 + T + 1.36e3T^{2} \) |
| 41 | \( 1 + 40T + 1.68e3T^{2} \) |
| 43 | \( 1 + 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 85iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 53iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 79T + 3.72e3T^{2} \) |
| 67 | \( 1 - 11iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 48iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 143T + 5.32e3T^{2} \) |
| 79 | \( 1 + 35iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 97T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88T + 9.40e3T^{2} \) |
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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
−8.702063773669085795394885655405, −8.296238855552437280075706780499, −7.06347801162689535995625635487, −6.49904914379316730823489490103, −5.90885972257986475775909418310, −5.27495801789066198970121805617, −3.54053432155495119843057554194, −2.22724778417055994938581691713, −1.80768626290080804621631050135, −0.73542588193810707843183435653,
1.47787924745714228287174659120, 2.67761028859706087989820194116, 3.64811339888607467423058071506, 4.63721334914154721461737840511, 5.43496385337386292322839302535, 5.96315856884367679319921801699, 6.88219835375703289610344491229, 8.384310049714808237554195919605, 9.116116238778498487166477031995, 9.586977667026604313988344631153
