| L(s) = 1 | − 2.72·2-s − 3-s + 5.45·4-s + 1.75·5-s + 2.72·6-s − 9.41·8-s + 9-s − 4.79·10-s + 0.440·11-s − 5.45·12-s + 0.286·13-s − 1.75·15-s + 14.8·16-s − 2.26·17-s − 2.72·18-s + 0.502·19-s + 9.58·20-s − 1.20·22-s + 23-s + 9.41·24-s − 1.90·25-s − 0.781·26-s − 27-s − 8.85·29-s + 4.79·30-s − 2.45·31-s − 21.5·32-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 0.577·3-s + 2.72·4-s + 0.786·5-s + 1.11·6-s − 3.33·8-s + 0.333·9-s − 1.51·10-s + 0.132·11-s − 1.57·12-s + 0.0793·13-s − 0.453·15-s + 3.70·16-s − 0.549·17-s − 0.643·18-s + 0.115·19-s + 2.14·20-s − 0.256·22-s + 0.208·23-s + 1.92·24-s − 0.381·25-s − 0.153·26-s − 0.192·27-s − 1.64·29-s + 0.876·30-s − 0.440·31-s − 3.81·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6247409592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6247409592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 11 | \( 1 - 0.440T + 11T^{2} \) |
| 13 | \( 1 - 0.286T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 - 0.502T + 19T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 + 2.45T + 31T^{2} \) |
| 37 | \( 1 - 5.08T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4.20T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 1.94T + 73T^{2} \) |
| 79 | \( 1 - 7.59T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 1.36T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.847475714464884416871673636513, −7.80486870297479529218194062982, −7.42930331148037265991639517003, −6.35696604119722237352148165164, −6.12740546448002100002881648170, −5.14054999298691651755992417417, −3.67760914702859728329268479899, −2.39829472333666125493842364060, −1.74861802737681958608353567394, −0.64462393111849106133353808901,
0.64462393111849106133353808901, 1.74861802737681958608353567394, 2.39829472333666125493842364060, 3.67760914702859728329268479899, 5.14054999298691651755992417417, 6.12740546448002100002881648170, 6.35696604119722237352148165164, 7.42930331148037265991639517003, 7.80486870297479529218194062982, 8.847475714464884416871673636513
