L(s) = 1 | + (−0.707 − 0.707i)3-s + (2.68 − 2.68i)5-s − 2.15i·7-s + 1.00i·9-s + (−1.79 + 1.79i)11-s + (−1.38 − 1.38i)13-s − 3.79·15-s − 0.224·17-s + (−0.158 − 0.158i)19-s + (−1.52 + 1.52i)21-s − 2.82i·23-s − 9.42i·25-s + (0.707 − 0.707i)27-s + (1.85 + 1.85i)29-s + 1.84·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (1.20 − 1.20i)5-s − 0.816i·7-s + 0.333i·9-s + (−0.542 + 0.542i)11-s + (−0.383 − 0.383i)13-s − 0.980·15-s − 0.0545·17-s + (−0.0364 − 0.0364i)19-s + (−0.333 + 0.333i)21-s − 0.589i·23-s − 1.88i·25-s + (0.136 − 0.136i)27-s + (0.344 + 0.344i)29-s + 0.330·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0734 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.902111 - 0.970966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.902111 - 0.970966i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-2.68 + 2.68i)T - 5iT^{2} \) |
| 7 | \( 1 + 2.15iT - 7T^{2} \) |
| 11 | \( 1 + (1.79 - 1.79i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.38 + 1.38i)T + 13iT^{2} \) |
| 17 | \( 1 + 0.224T + 17T^{2} \) |
| 19 | \( 1 + (0.158 + 0.158i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-1.85 - 1.85i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + (-3.66 + 3.66i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.88iT - 41T^{2} \) |
| 43 | \( 1 + (-7.75 + 7.75i)T - 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (7.51 - 7.51i)T - 53iT^{2} \) |
| 59 | \( 1 + (4 - 4i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.98 + 5.98i)T + 61iT^{2} \) |
| 67 | \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.31iT - 71T^{2} \) |
| 73 | \( 1 - 5.97iT - 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.42iT - 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−10.95476104714711540846630720702, −10.15157676814099578615081062257, −9.405532104059584711976019731006, −8.303455667157326429618446391549, −7.31730929443987066891645652572, −6.19630872933798323770797005438, −5.24086991304422973994705758757, −4.44246871174000371768462021508, −2.35500381271730125670881156796, −0.965250353819026023963088958118,
2.19916131676402039144251626707, 3.20774815438433283975246711094, 4.95366614377944373807127284371, 5.94104641587814872471612735941, 6.48901729917053069225430063772, 7.79338749919178174043459799152, 9.172521754341003060663714339838, 9.795547437842994024601492476899, 10.67726677042936964122037261125, 11.33209604281755452674868648119