| L(s) = 1 | − 32i·2-s + 243i·3-s − 1.02e3·4-s + 7.77e3·6-s − 3.29e4i·7-s + 3.27e4i·8-s − 5.90e4·9-s − 7.58e5·11-s − 2.48e5i·12-s − 2.48e6i·13-s − 1.05e6·14-s + 1.04e6·16-s − 8.29e6i·17-s + 1.88e6i·18-s + 1.08e7·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.740i·7-s + 0.353i·8-s − 0.333·9-s − 1.42·11-s − 0.288i·12-s − 1.85i·13-s − 0.523·14-s + 0.250·16-s − 1.41i·17-s + 0.235i·18-s + 1.00·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.3021273914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3021273914\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32iT \) |
| 3 | \( 1 - 243iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.29e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 7.58e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.48e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 8.29e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.08e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.05e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 2.88e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.50e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.19e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 + 3.68e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.20e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 2.76e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 2.68e8iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 1.67e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 7.78e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.87e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 8.34e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.96e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 5.87e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 8.49e9iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 7.55e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.23e10iT - 7.15e21T^{2} \) |
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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
−10.25462763856193838452369635168, −9.689979934952496225457708947313, −8.204602801633709920419267474473, −7.42021100627749559380665908724, −5.48405884502030764396001209573, −4.86593528613189096714458909660, −3.35176976738303767713859556256, −2.73563108614382818720181328806, −0.917971123467026450169090864106, −0.07507037311913935614395960628,
1.55533609166997388135542607637, 2.69314968336604762643283510706, 4.31450033951171030666217761693, 5.52353550549315358293263465411, 6.39496349839111657848184456798, 7.47567764577310407171898212339, 8.404634486803479975029446375285, 9.283819844701245658057425101401, 10.57694699427357697665343155541, 11.82875053180859129722485229132
