| L(s) = 1 | + (15.3 − 22.1i)3-s − 437. i·7-s + (−254. − 683. i)9-s − 1.89e3i·11-s − 2.67e3i·13-s − 4.52e3·17-s + 9.50e3·19-s + (−9.69e3 − 6.73e3i)21-s + 549.·23-s + (−1.90e4 − 4.86e3i)27-s + 4.33e4i·29-s + 2.95e4·31-s + (−4.21e4 − 2.92e4i)33-s + 4.57e4i·37-s + (−5.92e4 − 4.11e4i)39-s + ⋯ |
| L(s) = 1 | + (0.570 − 0.821i)3-s − 1.27i·7-s + (−0.349 − 0.936i)9-s − 1.42i·11-s − 1.21i·13-s − 0.920·17-s + 1.38·19-s + (−1.04 − 0.726i)21-s + 0.0451·23-s + (−0.968 − 0.247i)27-s + 1.77i·29-s + 0.993·31-s + (−1.17 − 0.813i)33-s + 0.904i·37-s + (−0.998 − 0.693i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.089493215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089493215\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-15.3 + 22.1i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 437. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.89e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.67e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 4.52e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 9.50e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 549.T + 1.48e8T^{2} \) |
| 29 | \( 1 - 4.33e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.95e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 4.57e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.14e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 2.49e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.82e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.74e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 6.78e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.75e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.60e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 2.22e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.20e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 5.25e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 7.90e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.12e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 6.24e5iT - 8.32e11T^{2} \) |
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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
−10.34070038676483504137118590367, −9.056536241249748596879056362833, −8.187015549846621136387617967581, −7.38144825881165551709939407510, −6.50076294618229467348080654419, −5.29200987549327186156080431292, −3.64964987921901694019937954142, −2.93308097355111182606165965137, −1.19188239752363578298802666343, −0.48643960344778992303994324394,
1.92500758071220742688433178250, 2.71071543449832522610359419875, 4.23274646415613275658594817847, 4.95285366994311772986669251231, 6.23035518662011879496059122074, 7.50518316068805303017242918692, 8.569407268361754144296687309654, 9.519477258302580243801583743760, 9.791022105424348749101566100728, 11.34807668706962348642560561068
