| L(s) = 1 | − 1.41i·2-s + (−0.0610 − 2.99i)3-s − 2.00·4-s − 5.31i·5-s + (−4.24 + 0.0863i)6-s − 5.79·7-s + 2.82i·8-s + (−8.99 + 0.366i)9-s − 7.52·10-s + 21.2i·11-s + (0.122 + 5.99i)12-s − 10.1·13-s + 8.20i·14-s + (−15.9 + 0.324i)15-s + 4.00·16-s − 17.1i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.0203 − 0.999i)3-s − 0.500·4-s − 1.06i·5-s + (−0.706 + 0.0143i)6-s − 0.828·7-s + 0.353i·8-s + (−0.999 + 0.0406i)9-s − 0.752·10-s + 1.93i·11-s + (0.0101 + 0.499i)12-s − 0.780·13-s + 0.585i·14-s + (−1.06 + 0.0216i)15-s + 0.250·16-s − 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0203 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0203 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.237075 + 0.241950i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.237075 + 0.241950i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (0.0610 + 2.99i)T \) |
| 59 | \( 1 + 7.68iT \) |
| good | 5 | \( 1 + 5.31iT - 25T^{2} \) |
| 7 | \( 1 + 5.79T + 49T^{2} \) |
| 11 | \( 1 - 21.2iT - 121T^{2} \) |
| 13 | \( 1 + 10.1T + 169T^{2} \) |
| 17 | \( 1 + 17.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.2T + 361T^{2} \) |
| 23 | \( 1 + 23.1iT - 529T^{2} \) |
| 29 | \( 1 - 3.84iT - 841T^{2} \) |
| 31 | \( 1 + 42.0T + 961T^{2} \) |
| 37 | \( 1 + 60.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 35.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 31.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 24.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 95.9iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 75.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.09T + 4.48e3T^{2} \) |
| 71 | \( 1 + 67.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 112.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 43.1T + 9.40e3T^{2} \) |
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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
−10.56264633146360111003044004311, −9.446262812370712728554536366074, −9.039218278850806375221115470427, −7.57656883144809798572328533737, −6.97216751230011956075737837742, −5.40925530906176353976589105513, −4.55361971848396871656200097164, −2.84613167418882780799026762848, −1.65697360001711978635470767573, −0.15094640850202251127986483311,
3.14920328328880376752570898057, 3.62990353775936348596123478982, 5.37244957857904215995400642183, 6.04638085014474741783047494976, 7.07162227768136559217597609404, 8.265051553584879386348743143112, 9.207049082001235484714528725986, 10.05695583835761787012545007775, 10.84507674681781605304964720850, 11.63921857728500779386050928113
