| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 2.37i·7-s + i·8-s − 9-s + 6.37·11-s − i·12-s − 2i·13-s + 2.37·14-s + 16-s − 6.74i·17-s + i·18-s + 6.37·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.896i·7-s + 0.353i·8-s − 0.333·9-s + 1.92·11-s − 0.288i·12-s − 0.554i·13-s + 0.634·14-s + 0.250·16-s − 1.63i·17-s + 0.235i·18-s + 1.46·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954283261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954283261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2.37iT - 7T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6.74iT - 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 + 2.37iT - 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 6.37iT - 43T^{2} \) |
| 47 | \( 1 + 4.74iT - 47T^{2} \) |
| 53 | \( 1 + 4.37iT - 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 0.744iT - 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 - 9.11iT - 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.564437924577951122625821265536, −7.47748480525144534424756675145, −6.73790620579056285521104059695, −5.69066009269952377787470003935, −5.22119626429661962061792738224, −4.32949750044530373399192064255, −3.48145439728809235187050913172, −2.88776746376828554023179361215, −1.80860740376077973947458704256, −0.63031658458938657219158967106,
1.12829709129633719385920323869, 1.65056518463477491577099012239, 3.44918238102332009410662083718, 3.84865571826920472773123663969, 4.75532876547109067257743985479, 5.76850325262519766482429540379, 6.47406582864686243003338745253, 6.90990231477663024138599582273, 7.56926648833517957962072345348, 8.309355448594299067081099935968
