| L(s) = 1 | + (−1.79 − 2.40i)3-s + 10.2i·7-s + (−2.53 + 8.63i)9-s − 8.19i·11-s − 13.5i·13-s − 15.4·17-s + 25.4·19-s + (24.5 − 18.3i)21-s − 17.9·23-s + (25.2 − 9.44i)27-s − 42.0i·29-s + 38.4·31-s + (−19.6 + 14.7i)33-s − 11.8i·37-s + (−32.6 + 24.4i)39-s + ⋯ |
| L(s) = 1 | + (−0.599 − 0.800i)3-s + 1.45i·7-s + (−0.281 + 0.959i)9-s − 0.744i·11-s − 1.04i·13-s − 0.910·17-s + 1.34·19-s + (1.16 − 0.874i)21-s − 0.778·23-s + (0.936 − 0.349i)27-s − 1.44i·29-s + 1.24·31-s + (−0.596 + 0.446i)33-s − 0.319i·37-s + (−0.836 + 0.626i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9253659184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9253659184\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.79 + 2.40i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10.2iT - 49T^{2} \) |
| 11 | \( 1 + 8.19iT - 121T^{2} \) |
| 13 | \( 1 + 13.5iT - 169T^{2} \) |
| 17 | \( 1 + 15.4T + 289T^{2} \) |
| 19 | \( 1 - 25.4T + 361T^{2} \) |
| 23 | \( 1 + 17.9T + 529T^{2} \) |
| 29 | \( 1 + 42.0iT - 841T^{2} \) |
| 31 | \( 1 - 38.4T + 961T^{2} \) |
| 37 | \( 1 + 11.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 54.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 43.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 82.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 45.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 68.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 44.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 11.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 144.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 63.9iT - 9.40e3T^{2} \) |
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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
−10.30552465115143375897367303950, −9.180818322456125000201763717688, −8.285554488949893980289865473460, −7.62013070586125093827871082512, −6.25034939864268480852154382693, −5.81070297316767521449054459982, −4.89988978921191517871141725755, −3.09150706665329244063771098113, −2.06859726957495963987396740238, −0.40206772980979928299462597967,
1.29839142677623791449617422800, 3.28552207917340224147781512350, 4.40433147168054085088728339367, 4.82808359156178385862924092099, 6.38092506692478354926119723406, 6.96798729345837943775645487003, 8.051354750358626238846999335033, 9.426652101842766682083946703456, 9.819678903596226685382011305167, 10.75747499483356640009333095656
