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
−11.33209604281755452674868648119, −10.67726677042936964122037261125, −9.795547437842994024601492476899, −9.172521754341003060663714339838, −7.79338749919178174043459799152, −6.48901729917053069225430063772, −5.94104641587814872471612735941, −4.95366614377944373807127284371, −3.20774815438433283975246711094, −2.19916131676402039144251626707,
0.965250353819026023963088958118, 2.35500381271730125670881156796, 4.44246871174000371768462021508, 5.24086991304422973994705758757, 6.19630872933798323770797005438, 7.31730929443987066891645652572, 8.303455667157326429618446391549, 9.405532104059584711976019731006, 10.15157676814099578615081062257, 10.95476104714711540846630720702