L(s) = 1 | + (−1.92 − 0.534i)2-s + 1.73·3-s + (3.42 + 2.05i)4-s + (−3.33 − 0.925i)6-s + 11.9·7-s + (−5.51 − 5.79i)8-s + 2.99·9-s + 14.5i·11-s + (5.94 + 3.56i)12-s − 22.4i·13-s + (−23.0 − 6.39i)14-s + (7.52 + 14.1i)16-s + 12.6i·17-s + (−5.78 − 1.60i)18-s + 8.76i·19-s + ⋯ |
L(s) = 1 | + (−0.963 − 0.267i)2-s + 0.577·3-s + (0.857 + 0.514i)4-s + (−0.556 − 0.154i)6-s + 1.71·7-s + (−0.688 − 0.724i)8-s + 0.333·9-s + 1.32i·11-s + (0.495 + 0.297i)12-s − 1.72i·13-s + (−1.64 − 0.456i)14-s + (0.470 + 0.882i)16-s + 0.746i·17-s + (−0.321 − 0.0890i)18-s + 0.461i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.56739 - 0.0603677i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56739 - 0.0603677i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.92 + 0.534i)T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 11.9T + 49T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 + 22.4iT - 169T^{2} \) |
| 17 | \( 1 - 12.6iT - 289T^{2} \) |
| 19 | \( 1 - 8.76iT - 361T^{2} \) |
| 23 | \( 1 - 4.99T + 529T^{2} \) |
| 29 | \( 1 + 2.74T + 841T^{2} \) |
| 31 | \( 1 + 16.3iT - 961T^{2} \) |
| 37 | \( 1 - 32.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 16.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 48.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 94.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 43.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 61.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 39.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 10.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 140.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 54.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 14.1iT - 9.40e3T^{2} \) |
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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
−11.27383065134341913572508097424, −10.44802006674984044548459559028, −9.716853277317340472186795129484, −8.382854527137161843355214253304, −8.006621536580683919091589751897, −7.16529788961955212851640649683, −5.49249770560154358284287965208, −4.10274392264708794607548257577, −2.49620246120529837286403489245, −1.37197105480943763882842600594,
1.25532565454947277615486933516, 2.51982794840729700792226930622, 4.38984856074442004550955772406, 5.68058539144558019273891752661, 7.04087904109109930534860613909, 7.82096683463357518172823616852, 8.866082252002472469953954529138, 9.133704690110111997234554454723, 10.72883881531666937666654096315, 11.26989329534391553505232546416