| L(s) = 1 | − 0.630·3-s − 3.98i·7-s − 8.60·9-s − 5.54·11-s − 23.1·13-s + 16.5i·17-s + (−11.6 − 15.0i)19-s + 2.51i·21-s − 6.86i·23-s + 11.0·27-s + 37.0i·29-s − 11.1i·31-s + 3.49·33-s + 66.6·37-s + 14.5·39-s + ⋯ |
| L(s) = 1 | − 0.210·3-s − 0.569i·7-s − 0.955·9-s − 0.504·11-s − 1.77·13-s + 0.970i·17-s + (−0.613 − 0.790i)19-s + 0.119i·21-s − 0.298i·23-s + 0.410·27-s + 1.27i·29-s − 0.358i·31-s + 0.105·33-s + 1.80·37-s + 0.373·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9879259273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9879259273\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (11.6 + 15.0i)T \) |
| good | 3 | \( 1 + 0.630T + 9T^{2} \) |
| 7 | \( 1 + 3.98iT - 49T^{2} \) |
| 11 | \( 1 + 5.54T + 121T^{2} \) |
| 13 | \( 1 + 23.1T + 169T^{2} \) |
| 17 | \( 1 - 16.5iT - 289T^{2} \) |
| 23 | \( 1 + 6.86iT - 529T^{2} \) |
| 29 | \( 1 - 37.0iT - 841T^{2} \) |
| 31 | \( 1 + 11.1iT - 961T^{2} \) |
| 37 | \( 1 - 66.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 14.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 63.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 36.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 27.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 19.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 66.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.0T + 9.40e3T^{2} \) |
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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
−9.064030662183974935392343334567, −8.212532601286011715054530064169, −7.50003400044118608420821394555, −6.72334345154423412346138491771, −5.82546707836842299805065730887, −4.97577908431172840520002109517, −4.25723689938387322234481619290, −2.98185803214964953440908245553, −2.22160167332901569553094112376, −0.57587907402724089672612033744,
0.45365619851687038671918355142, 2.34354884425347074011807921251, 2.72374362463415020461686905742, 4.14431555317179813829848710775, 5.16054255320818601749061634494, 5.62799468214111685978111927236, 6.58781036063099053187872843229, 7.55503513481527524406260897531, 8.145701224189949769914707444798, 9.060885522569624664810777281580
