L(s) = 1 | + (64 − 64i)2-s + (3.06e3 + 3.06e3i)3-s − 8.19e3i·4-s + (2.06e4 − 7.53e4i)5-s + 3.91e5·6-s + (5.86e5 − 5.86e5i)7-s + (−5.24e5 − 5.24e5i)8-s + 1.39e7i·9-s + (−3.49e6 − 6.14e6i)10-s + 1.03e7·11-s + (2.50e7 − 2.50e7i)12-s + (2.36e7 + 2.36e7i)13-s − 7.50e7i·14-s + (2.93e8 − 1.67e8i)15-s − 6.71e7·16-s + (−2.68e8 + 2.68e8i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (1.39 + 1.39i)3-s − 0.5i·4-s + (0.264 − 0.964i)5-s + 1.39·6-s + (0.711 − 0.711i)7-s + (−0.250 − 0.250i)8-s + 2.91i·9-s + (−0.349 − 0.614i)10-s + 0.531·11-s + (0.699 − 0.699i)12-s + (0.376 + 0.376i)13-s − 0.711i·14-s + (1.71 − 0.979i)15-s − 0.250·16-s + (−0.653 + 0.653i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0357i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.999 + 0.0357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(3.88437 - 0.0695276i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.88437 - 0.0695276i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-64 + 64i)T \) |
| 5 | \( 1 + (-2.06e4 + 7.53e4i)T \) |
good | 3 | \( 1 + (-3.06e3 - 3.06e3i)T + 4.78e6iT^{2} \) |
| 7 | \( 1 + (-5.86e5 + 5.86e5i)T - 6.78e11iT^{2} \) |
| 11 | \( 1 - 1.03e7T + 3.79e14T^{2} \) |
| 13 | \( 1 + (-2.36e7 - 2.36e7i)T + 3.93e15iT^{2} \) |
| 17 | \( 1 + (2.68e8 - 2.68e8i)T - 1.68e17iT^{2} \) |
| 19 | \( 1 + 4.55e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + (3.95e7 + 3.95e7i)T + 1.15e19iT^{2} \) |
| 29 | \( 1 + 1.04e10iT - 2.97e20T^{2} \) |
| 31 | \( 1 + 1.10e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + (7.04e9 - 7.04e9i)T - 9.01e21iT^{2} \) |
| 41 | \( 1 + 2.23e10T + 3.79e22T^{2} \) |
| 43 | \( 1 + (2.16e11 + 2.16e11i)T + 7.38e22iT^{2} \) |
| 47 | \( 1 + (2.30e11 - 2.30e11i)T - 2.56e23iT^{2} \) |
| 53 | \( 1 + (3.91e11 + 3.91e11i)T + 1.37e24iT^{2} \) |
| 59 | \( 1 + 4.44e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 2.34e12T + 9.87e24T^{2} \) |
| 67 | \( 1 + (5.17e12 - 5.17e12i)T - 3.67e25iT^{2} \) |
| 71 | \( 1 - 1.06e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + (-7.66e11 - 7.66e11i)T + 1.22e26iT^{2} \) |
| 79 | \( 1 - 2.49e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 + (-2.02e13 - 2.02e13i)T + 7.36e26iT^{2} \) |
| 89 | \( 1 - 5.70e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-1.04e14 + 1.04e14i)T - 6.52e27iT^{2} \) |
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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
−16.94994207245584178461673327876, −15.56912958252111676777830038497, −14.28277945336072557532676146312, −13.37162962425370269565057657651, −10.98597650481629184138030515979, −9.559114811688637250785244975759, −8.392840634234633617818059867675, −4.78773722628905976724535508570, −3.86545123183683878876312190212, −1.84730347520249692770659888793,
1.87983804809525247919443860652, 3.20820385711959862945903493130, 6.32893321261473851469671233511, 7.58295278099560669685996530728, 8.904115665503826775200664331824, 11.85692853294414018969821010435, 13.35814152920104400714906818149, 14.39416055557850558407962705646, 15.12868924290956385340170614573, 17.89890983410770865065425035481