| L(s) = 1 | − 0.329·2-s + 3-s − 1.89·4-s + 2.73·5-s − 0.329·6-s + 1.28·8-s + 9-s − 0.902·10-s − 2.50·11-s − 1.89·12-s − 1.48·13-s + 2.73·15-s + 3.35·16-s − 0.902·17-s − 0.329·18-s − 2.50·19-s − 5.17·20-s + 0.825·22-s + 23-s + 1.28·24-s + 2.48·25-s + 0.489·26-s + 27-s + 6.68·29-s − 0.902·30-s − 1.09·31-s − 3.67·32-s + ⋯ |
| L(s) = 1 | − 0.233·2-s + 0.577·3-s − 0.945·4-s + 1.22·5-s − 0.134·6-s + 0.453·8-s + 0.333·9-s − 0.285·10-s − 0.755·11-s − 0.545·12-s − 0.411·13-s + 0.706·15-s + 0.839·16-s − 0.218·17-s − 0.0777·18-s − 0.574·19-s − 1.15·20-s + 0.176·22-s + 0.208·23-s + 0.261·24-s + 0.497·25-s + 0.0960·26-s + 0.192·27-s + 1.24·29-s − 0.164·30-s − 0.197·31-s − 0.649·32-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
| \(L(1)\) |
\(\approx\) |
\(1.949583000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.949583000\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.329T + 2T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 + 0.902T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 - 5.83T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 - 7.32T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 0.362T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.59T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
| 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
−8.648845447849085772181750684315, −8.088952948073867262480643504494, −7.28995337365300565086589262814, −6.28920396041634871778143616391, −5.53601479610653638086036111760, −4.78130119987258221374666823300, −4.04430546859387324448689388063, −2.80143040896507151707403441487, −2.13373336035446610503052003188, −0.852861991991762337128025979377,
0.852861991991762337128025979377, 2.13373336035446610503052003188, 2.80143040896507151707403441487, 4.04430546859387324448689388063, 4.78130119987258221374666823300, 5.53601479610653638086036111760, 6.28920396041634871778143616391, 7.28995337365300565086589262814, 8.088952948073867262480643504494, 8.648845447849085772181750684315
