| L(s) = 1 | + (−2.86 + 0.888i)3-s − 8.59i·5-s − 10.9·7-s + (7.41 − 5.09i)9-s − 2.75i·11-s − 4.43·13-s + (7.63 + 24.6i)15-s + 25.4i·17-s + 17.5·19-s + (31.3 − 9.71i)21-s + 17.5i·23-s − 48.8·25-s + (−16.7 + 21.1i)27-s − 19.6i·29-s − 2.58·31-s + ⋯ |
| L(s) = 1 | + (−0.955 + 0.296i)3-s − 1.71i·5-s − 1.56·7-s + (0.824 − 0.565i)9-s − 0.250i·11-s − 0.341·13-s + (0.509 + 1.64i)15-s + 1.49i·17-s + 0.923·19-s + (1.49 − 0.462i)21-s + 0.762i·23-s − 1.95·25-s + (−0.619 + 0.784i)27-s − 0.676i·29-s − 0.0833·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.149173 + 0.202460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149173 + 0.202460i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.86 - 0.888i)T \) |
| good | 5 | \( 1 + 8.59iT - 25T^{2} \) |
| 7 | \( 1 + 10.9T + 49T^{2} \) |
| 11 | \( 1 + 2.75iT - 121T^{2} \) |
| 13 | \( 1 + 4.43T + 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 17.5T + 361T^{2} \) |
| 23 | \( 1 - 17.5iT - 529T^{2} \) |
| 29 | \( 1 + 19.6iT - 841T^{2} \) |
| 31 | \( 1 + 2.58T + 961T^{2} \) |
| 37 | \( 1 - 7.73T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 17.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 69.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 50.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 32.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 48.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 22.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 97.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 59.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 110. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 55.1T + 9.40e3T^{2} \) |
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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
−11.64560498922450396508893507756, −10.26856055207939955959154478686, −9.627674268203987328485342584417, −8.910417409310339167130250106276, −7.66188720932261521725495879727, −6.27646927628044957308796188703, −5.66512447727987898811713482805, −4.58878539366391634238581197189, −3.55135948872783259554338886855, −1.20821677972574415288188597650,
0.13909824807290844820878247311, 2.54866881538509597707144581350, 3.50518323985027702640348836970, 5.18296081711424988068580633690, 6.36510126609632291401918086481, 6.89310643460687930627594326731, 7.48460026869244868911691884230, 9.463338975366297676299628881505, 10.07152598701529546608855629198, 10.82070952369078634594762846731
