| L(s) = 1 | + (−0.399 + 1.35i)2-s + (2.51 − 0.630i)3-s + (−1.68 − 1.08i)4-s + (2.55 + 1.20i)5-s + (−0.149 + 3.66i)6-s + (−0.0344 − 0.113i)7-s + (2.14 − 1.84i)8-s + (3.28 − 1.75i)9-s + (−2.65 + 2.97i)10-s + (−5.03 + 0.746i)11-s + (−4.91 − 1.66i)12-s + (1.80 − 5.05i)13-s + (0.167 − 0.00137i)14-s + (7.17 + 1.42i)15-s + (1.65 + 3.64i)16-s + (−4.56 + 0.908i)17-s + ⋯ |
| L(s) = 1 | + (−0.282 + 0.959i)2-s + (1.45 − 0.363i)3-s + (−0.840 − 0.541i)4-s + (1.14 + 0.539i)5-s + (−0.0612 + 1.49i)6-s + (−0.0130 − 0.0429i)7-s + (0.757 − 0.653i)8-s + (1.09 − 0.585i)9-s + (−0.839 + 0.941i)10-s + (−1.51 + 0.225i)11-s + (−1.41 − 0.481i)12-s + (0.501 − 1.40i)13-s + (0.0448 − 0.000366i)14-s + (1.85 + 0.368i)15-s + (0.412 + 0.910i)16-s + (−1.10 + 0.220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57263 + 0.735944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57263 + 0.735944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.399 - 1.35i)T \) |
| good | 3 | \( 1 + (-2.51 + 0.630i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-2.55 - 1.20i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.0344 + 0.113i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (5.03 - 0.746i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-1.80 + 5.05i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (4.56 - 0.908i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-5.03 - 4.56i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (3.35 - 0.330i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 1.96i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (5.35 + 2.21i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.258 + 5.26i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (3.74 - 4.56i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (2.09 - 8.34i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-2.41 - 1.61i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-3.76 + 5.07i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (2.43 + 6.81i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-1.05 + 1.75i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (9.27 + 5.56i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-0.928 + 1.73i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-2.65 + 8.74i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (0.925 + 1.38i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.164 + 3.35i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-14.3 - 1.41i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (2.28 - 5.51i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−12.91963485129306471962043885482, −10.58987215551329350147490912260, −10.01676698695955343089724570692, −9.089176872155309398744495966218, −8.001133261223538676368922237182, −7.59937911342456583209473154246, −6.22092836554647115199697437324, −5.29438893255237236631424479400, −3.36813134666895143720518733011, −2.05298442373010952961736196498,
1.95842080301491631728961092241, 2.78793268431713505506394589091, 4.21643034242914667398403309376, 5.36113404460286863661453371667, 7.32338538497829577378833865406, 8.659351638009525120472039289321, 8.985929069730289420306090762882, 9.794924879373048835537822292765, 10.65165821524318811327734291695, 11.84212355147385030700040061498
