| L(s) = 1 | + 2.60·3-s − 2.23i·5-s + 8.05i·7-s − 2.22·9-s + 5.14i·11-s − 6.63·13-s − 5.82i·15-s − 10.7i·17-s − 9.70i·19-s + 20.9i·21-s + (−16.6 − 15.9i)23-s − 5.00·25-s − 29.2·27-s − 9.52·29-s + 10.0·31-s + ⋯ |
| L(s) = 1 | + 0.867·3-s − 0.447i·5-s + 1.15i·7-s − 0.247·9-s + 0.468i·11-s − 0.510·13-s − 0.388i·15-s − 0.635i·17-s − 0.510i·19-s + 0.998i·21-s + (−0.722 − 0.691i)23-s − 0.200·25-s − 1.08·27-s − 0.328·29-s + 0.324·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8135708708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8135708708\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (16.6 + 15.9i)T \) |
| good | 3 | \( 1 - 2.60T + 9T^{2} \) |
| 7 | \( 1 - 8.05iT - 49T^{2} \) |
| 11 | \( 1 - 5.14iT - 121T^{2} \) |
| 13 | \( 1 + 6.63T + 169T^{2} \) |
| 17 | \( 1 + 10.7iT - 289T^{2} \) |
| 19 | \( 1 + 9.70iT - 361T^{2} \) |
| 29 | \( 1 + 9.52T + 841T^{2} \) |
| 31 | \( 1 - 10.0T + 961T^{2} \) |
| 37 | \( 1 + 54.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 47.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 29.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 86.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 4.11T + 3.48e3T^{2} \) |
| 61 | \( 1 + 17.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 49.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 42.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 52.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 110. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 138. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 163. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 90.4iT - 9.40e3T^{2} \) |
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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
−8.771204974926013551277221689844, −8.194369763635550544800873094896, −7.38862967616775135076682425687, −6.36831329829278363057910822949, −5.41983973826718741092917436609, −4.75155912808179140806546997577, −3.58438527726883109694818500350, −2.55113509426416046864608780393, −2.01041619858425681819887553660, −0.17119675712263131078427052262,
1.41089095484184255939031043728, 2.59180227682824301438941407063, 3.52075866584104621994546246123, 4.07131668718030789782450564991, 5.34340494565392691008750437443, 6.29151951440309525348808023272, 7.14112207849680786154867289613, 7.917584596011711030826361081198, 8.369301653000032104347041155791, 9.394842709602951768578856357850
