L(s) = 1 | + 1.76·5-s + 12.0i·7-s + 13.3i·11-s − 8.15·13-s + 4.15·17-s − 8.87i·19-s + 33.5i·23-s − 21.8·25-s − 36.4·29-s + 0.590i·31-s + 21.1i·35-s + 69.8·37-s − 57.5·41-s − 23.5i·43-s − 24.4i·47-s + ⋯ |
L(s) = 1 | + 0.352·5-s + 1.71i·7-s + 1.21i·11-s − 0.627·13-s + 0.244·17-s − 0.466i·19-s + 1.46i·23-s − 0.875·25-s − 1.25·29-s + 0.0190i·31-s + 0.605i·35-s + 1.88·37-s − 1.40·41-s − 0.548i·43-s − 0.520i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9848993190\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9848993190\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.76T + 25T^{2} \) |
| 7 | \( 1 - 12.0iT - 49T^{2} \) |
| 11 | \( 1 - 13.3iT - 121T^{2} \) |
| 13 | \( 1 + 8.15T + 169T^{2} \) |
| 17 | \( 1 - 4.15T + 289T^{2} \) |
| 19 | \( 1 + 8.87iT - 361T^{2} \) |
| 23 | \( 1 - 33.5iT - 529T^{2} \) |
| 29 | \( 1 + 36.4T + 841T^{2} \) |
| 31 | \( 1 - 0.590iT - 961T^{2} \) |
| 37 | \( 1 - 69.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 57.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 23.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 24.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 29.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 55.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 11.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 99.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 71.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 36.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 108. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 100. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 159.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 24.0T + 9.40e3T^{2} \) |
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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
−9.641122446747003529256822723931, −8.920326067707773184997119885437, −7.914231181187276612984393608699, −7.22762494274044961408755549260, −6.17915329890528955440272306835, −5.44421268430056035639192758052, −4.85559263656494662865966362357, −3.55211241508641162860924306647, −2.36174429769374819848545060235, −1.83263926691029083521820779761,
0.25486387132689945346518882163, 1.27226576530927049700666884092, 2.68354661988684250820811216718, 3.77530794288705832378594862212, 4.41022488427163008378410570756, 5.57387821798299413938881380535, 6.33807606996670131392359953326, 7.21868353214996106844907932811, 7.88224203421637221489513178229, 8.649378042482834925541620909650